ANALYSIS OF FLUCTUATIONS IN THE EARLY UNIVERSEUSING FOKKER-PLANCK FORMALISM

Thesis submittedin partial fulfilment of the requirements

for the award of the Degree of

Doctor of Philosophy

in

Physics

Department of PhysicsCochin University of Science and Technology

Kochi-22, KeralaIndia

2001

DECLARATION

I hereby declare that the thesis titled Analysis of Fluctuations in the

Early Universe Using Fokker-Planck Formalism, submitted for the degree

of Doctor of Philosophy is a bonafide record of the research carried out by me

under the guidance of Prof. K Babu Joseph, in the Department of Physics,

Cochin University of Science and Technology, and that no part of it has been

included in any other thesis submitted previously for the award of any degree of

any university.

.1‘

Kochi-22 Siva umar CDate- t rid * '

CERTIFICATE

Certified that the thesis titled Analysis of Fluctuations in the Early Uni­

verse Using Fokker-Planck Formalism is a bonafide record of the research

carried out by Mr. Sivakumar C, under my supervision in the Department of

Physics, Cochin University of Science and Technology, in partial fulfilment of the

requirements for the award of the Degree of Doctor of Philosophy, and no part

of it has been included in any other thesis submitted previously for the award of

any degree of any university.

,\Z."’ .ea“-“" —Kochi-22 Prof. K Babu JosephDate- \ 1CC l (Supervising teacher)

ACKNOWLEDGEMENTS

It is a pleasure to express my profound and sincere gratitude to Prof. K Babu

Joseph for his invaluable guidance and encouragement throughout my career, first

as a post graduate student and later as a research student. His constant help and

vast knowledge helped me a lot during the period of my research.

I thank Prof. Elizabeth Mathai, Head, Department of Physics, for allowing

me to use the facilities in the department for carrying out my research.

I am extremely thankful to Dr. Moncy V John, St. Thomas college, Kozhencherry,

for his help and valuable suggestions in carrying out this work.

This work was carried out in the Department of Physics, Cochin University,

and all the faculty members, students and office staff of the department have

been very supportive to me in every respect and I thank them all.

I am thankful to Prof. M Sabir, Prof. V C Kuriakose and Dr. Ramesh Babu

T for their valuable suggestions and support.

My warm thanks to Taji, J ayadevan, Ganapathy, Sheeja, Shaju, Viji, Raviku­

mar, Vinoj, Sandhya, Minu, Manju, Sajith, Rajesh, Shaji, Aldrin, Fr. Tomy, Dr.

G Vinod and Dr. K P Satheesh for all their cooperation.

I would like to thank IUCAA, Pune, for the warm hospitality extended to me

during my visit. My special thanks are due to Prof. T Padmanabhan and Prof.

Ajith Kembhavi for their suggestions and discussions which became very helpful

in my research.

I gratefully acknowledge CSIR, New Delhi, for the award of a research fellow­

ship during the period 1996 to 2001.

Finally, I would like to thank my parents and other members of my family for

their constant affection and inspiration in the pursuit of my research work.

Dedicated to my parents

Contents

Preface

1 The1.1

1.2

1.3

1.4

2 The2.12.22.3

Standard Cosmological modelEinstein’s General Theory of Relativity

1.1.1 Contravariant and Covariant tensors1.1.2 Metric tensor1.1.3 Contraction and raising and lowering of indices1.1.4 Covariant differentiation and parallel transport1.1.5 Spacetime curvature and the Riemannian curvature tensor1.1.6 The Principle of Equivalence(PE)1.1.7 Einstein’s field equations1.1.8 Earlier solutionsThe Standard Cosmological model1.2.1 Robertson-Walker line element1.2.2 The Redshift1.2.3 Hubble’s law1.2.4 The Friedmann modelsStructure formation1.3.1 Linear perturbation theoryPredictions and problems of the standard model1.4.1 Hubb1e’s expansion and the redshift-apparent magnitude

(2 — m) relation1.4.2 Fractal distribution of galaxies1.4.3 The abundance of light nuclei1.4.4 The microwave background1.4.5 Other problems with the standard model and the inflation­

ary model

New approachIntroductionStochastic equation of stateStochastic approach to the standard model2.3.1 Stochastic Hubble parameter

30333839

40

4747495252

2.3.2 The Fokker-Planck equation2.4 Comparison with data and conclusions

3 Evolution of cosmological parameters in the new model3.1 Introduction3.2 Stochastic evolution of the cosmological parameters

3.2.1 Density parameter3.2.2 Scale factor and density parameter together as a two vari­

able Fokker-Planck problem3.3 Conclusion

4 Application of the stochastic approach to the generalized Chen­Wu type cosmological model4.1 Introduction4.2 Stochastic approach to the new model

4.2.1 Generalized Chen-Wu type cosmological model4.2.2 Evolution of the Hubble parameter in the new model4.2.3 PDF for H from observational data

4.3 Conclusions

5 Other problems with the standard model and discussions5.1 Introduction5.2 A stochastic evolution of density perturbations5.3 Discussions and conclusions

Bibliography

CONTENTS

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96

PREFACE

The work presented in this thesis has been carried out by the author at the

Department of Physics, Cochin University of Science and Technology during the

period 1996 to 2001.

Einstein’s General Theory of Relativity (GTR) laid the foundations of modern

theoretical cosmology, which assumes a central role in interpreting astrophysical

and cosmological data. It provides a frame-work which is free from the difficulties

of Newtonian gravity with respect to special theory of relativity and special the­

ory of relativity with respect to gravity. Unlike astronomy, cosmology is highly

speculative in nature and lacked observational evidence in early days. However,

during the past decade, the observational support to cosmology has tremendously

increased with the advent of the Hubble space telescope and the Cosmic Back­

ground Explorer (COBE) satellite. Yet, since we cannot experiment with the

cosmos, one can only resort to model making and check how much of the ob­

servational data agrees with the predictions of the model. Among the several

cosmological models that have been put forward, the standard model or the hot

big bang model originated by Friedrnann, provides the most successful approxi­

mation of the real universe, with maximum consistency with observations. This

model is based on following assumptions: (1) the geometry of space is such that

at large scales, it is describable by the mathematically simple, spatially symmet­

ric Friedmann-Robertson-Walker metric; (2) the energy-momentum tensor is that

which corresponds to a spatially homogeneous and isotropic perfect fluid compris­

ing matter which is either relativistic or non relativistic. This model predicts (1)

an early hot phase for the universe, a relic of which is the cosmic microwave back­

ground radiation (CMBR), (2) Hubble expansion and (3) the observed abundance

iii

of light nuclei in the universe.

However, there are certain problems in the standard model, some of which

are directly dependent upon the simplifying assumptions taken. The inflationary

model proposed in the early eighties, is a modification of the standard model,

which predicts an early exponential expansion for the universe, caused by the

potential energy of a scalar or inflation field. This model solves some of the prob­

lems in the standard model. The standard model predicts a linear deterministic

redshift-apparent magnitude (2 — m) relation (or a linear Hubble’s law). However

the Hubble diagram is a scatter diagram with no deterministic Hubble type rela­

tion clearly apparent, especially for higher 2 (in the early universe). The scatter is

also found to increase with redshift. Thus the behaviour of the Hubble parameter

(H is anomalous in those epochs. Also the uncertainty in the determination

of the true value of the Hubble parameter is one of the most intriguing issues

in the history of cosmology. In conventional cosmology, the peculiar velocities

induced by the observed density fluctuations, are the cause of the randomness

in the Hubble diagram. However, peculiar velocities are inadequate for high 2,

because density fluctuations are evolving phenomena. The observed fractal dis­

tribution of galactic clusters over large range of scales is another puzzle within

the standard model.

We begin the thesis with a review of basic elements of general theory of rel­

ativity (GTR) which forms the basis for the theoretical interpretation of the

observations in cosmology. The first chapter also discusses the standard model

in cosmology, namely the Friedmann model, its predictions and problems. We

have also made a brief discussion on fractals and inflation of the early universe

in the first chapter. In the second chapter we discuss the formulation of a new

approach to cosmology namely a stochastic approach. In this model, the dynam­

iv

ics of the early universe is described by a set of non-deterministic, Langevin type

equations and we derive the solutions using the Fokker—Planck formalism. Here

we demonstrate how the problems with the standard model, can be eliminated

by introducing the idea of stochastic fluctuations in the early universe. Many

recent observations indicate that the present universe may be approximated by

a many component fluid and we assume that only the total energy density is

conserved. This, in turn, leads to energy transfer between different components

of the cosmic fluid and fluctuations in such energy transfer can certainly induce

fluctuations in the mean to factor in the equation of state 12 = wp, resulting in

a fluctuating expansion rate for the universe. We also have made a comparison

between theoretical predictions and observations using the Type Ia supernovae

data in [25].

The third chapter discusses the stochastic evolution of the cosmological pa­

rameters in the early universe, using the new approach. The penultimate chap­

ter is about the refinements to be made in the present model, by means of a

new deterministic model [91]. The concluding chapter presents a discussion on

other problems with the conventional cosmology, like fractal correlation of galac­

tic distribution. We attempt an explanation for this problem using the stochastic

approach.

A part of these investigations has appeared in the form of the following papers

published/submitted/presented

1. C Sivakumar, M V John and K Babu Joseph, Pramana-J. Phys. 56, 477

(2001).

2. C Sivakumar, M V john and K Babu Joseph, Int. J. Mod. Phys.D,

submitted (2001).

3. M V John, C Sivakumar and K Babu Joseph, Pramana-J. Phys. submitted

(2001).

4. C Sivakumar and K Babu Joseph, to submit (2001).

5. Paper presented at the XIV DAE Sumposium on High energy physics

(Hyderabad, 2000) and submitted to the proceedings.

vi

Chapter 1

The Standard Cosmologicalmodel

Cosmology is the study of the origin and development of the universe, using the

tools of astronomy. Eventhough astronomy started as a branch of science dealing

with planets and stars, it now deals with objects which are very far away from

us, ie., light from such objects, take billions of years to reach us. On the other

hand, cosmology is mainly concerned with the extragalactic world, particularly

with the large-scale structure of the universe extending to distances of billions of

light years across. Towards the end of the third decade of the twentieth century,

Edwin Hubble came up with the discovery that the spectra of galaxies appear

to be shifted towards the red end of the spectrum, and that the shift in a given

galaxy is proportional to the distance of the galaxy from us (Doppler shift). This

striking observation actually laid the foundation for modern observational cos­

mology. In 1922, Alexander Friedmann had found model solutions of Einstein’s

equations of General Theory of Re1ativity(GTR), wherein the property of redshift

arose naturally. Friedmann’s cosmological models got observational support only

in 1929, when Hubble made that remarkable discovery. This model [1, 2, 3] forms

the basis of the currently standard picture in cosmology, which is a nearly homo­

geneous and isotropic expansion of the universe, according to GTR, that traces

1

back to a hot and dense state for the universe. And it was Steven Weinberg who

brought the phrase, “the standard model”, to cosmology from particle physics.

Since the standard model assumes general theory of relativity, let us begin with

a brief introduction to it, before going to the details of the Friedmann models.

1.1 Einstein’s General Theory of Relativity

General relativity [3-11], the modern theory of gravitation, provides a mathemat­

ical model of the physical world and laid the foundations of modern theoretical

cosmology. The problem with Newtonian theory of gravity is that it is a theory

of instantaneous action at a distance, and hence it fails at very large distances

(ie. on cosmological scales). In 1905, Einstein proposed the Special Theory of

Relativity(STR), which revolutionised the concepts of space, time and motion on

which the Newtonian laws were founded [10]. According to STR there is a limit

to the speed beyond which no interaction can propagate. This corresponds to the

speed of light which is a constant. Light can never be at rest relative to anything

and cannot be acted upon by any force. Then the question arises is, how is light

affected by gravitation? The problem with STR is the omnipresence of gravity.

No inertial frames and observers exist, which are the basis of STR. Einstein’s

GTR provides a framework that is free from the difficulties of Newtonian gravity

with respect to STR and STR with respect to gravity.

The most distinctive feature of gravity is its permanent character, ie., it is

an interaction which cannot be turned on or off at will. Einstein argued that,

because of its permanence, gravitation must be related to some intrinsic feature

of space and time [3, 11]. He identified this feature as the geometry of space

and time(or spacetime) and suggested that the geometry of space (spacetime) is

non-Euclidean, ie., spacetime is curved. Non-Euclidean character means that the

2

laws of Euclid are not valid. For example, geometry on the surface of a sphere

is non-Euclidean. If we define a straight line on the surface of the sphere as a

line of shortest distance between two points, it is easy to see that these lines are

great circles, and that any two of these intersect. Thus the laws of Euclid do not

hold here.

In GTR the use of scalars, vectors and tensors in non-Euclidean spacetime

is important because of the requirement of general covariance of physical laws,

independent of coordinate systems. Intrinsic properties of spacetime geometry

are described in terms of such geometrical objects, for example, the Riemann­

Christoffel curvature tensor.

1.1.1 Contravariant and Covariant tensors

In order to construct physical equations that are invariant under general coor­

dinate transformations, we have to know how the quantities described by the

equations behave under these transformations. The simplest transformation rule

is that of scalars, which are invariant under general coordinate transformations

(xi —) For vectors there are two kinds of transformation rules-contravariant

and covariant, and correspondingly we have contravariant and Covariant vectors

[3, 5]. The transformation law for a contravariant vector (components denoted

by Ai) is that, which under a coordinate transformation 2:‘ —> 1:" transform into

,. 83:,‘ ­A I = .A’,613 (1.1)

where Einstein’s summation convention is used. This follows the rules of partial

differentiation

d.'13i =

The transformation law of a covariant vector (B,-) isIFor example, if go is a scalar field, then 690/ 3:1? is a covariant vector,

8 Bzj 8—“'f. = —,i. (1.4)31: 1 Bx ‘ 81:1

This follows the covariant transformation law. Now we generalize these rules to

tensors. In fact a vector is a tensor of rank one and a scalar is a tensor of rank

zero. Now we express the transformation rules for second rank contravariant and

covariant tensors in the form,

I-- afrliT1] : j kl, ‘­3z’° 33:‘ ’ (1 O), 6 ’° 6 ‘T — I —’”.T,,,. (1.6)‘7 _ 82:" 627

It is straightforward to write the transformation equation for a mixed tensor

having two contravariant and two covariant indices (rank is four):

_ 327" 82:7 39:1’ BI‘? ,1T4’ ' (1-7’The distinction between contravariant and covariant tensors disappears in rect­

angular Cartesian coordinate systems. In general any vector have distinct sets of

contravariant and covariant components in an arbitrary coordinate system.

1.1.2 Metric tensor

In terms of the general coordinates, the line element of a non-Euclidean spacetime

is written as the quadratic form [5, 8]

3

ds2 = Z g,-kd:r‘da:" = g,~kda:‘d:1:" (1.8)i,k=0

The space-time coordinates are xi = z°,:r‘,:r2& 2:3 (:::° = t, with c = 1). gik is

called the metric tensor (from quotient law of tensors, gut is a second rank covari­

ant tensor so that dsz is an invariant). For the flat Minkowski spacetime of STR,

g,-,, has the diagonal form, ie., gik = {1, -1, -1, -1}. In general 91'}: is symmetric

in 71 and Is, hence there are at most ten linearly independent components. Eq.

(1.8) is said to define the space-time metric and we will assume the signature of

eq. (1.8) to be -2. For a curved spacetime, g.-k are coordinate dependent. How­

ever, sometimes the dependence is a purely coordinate effect and not intrinsic

to the curvature of space. For example, the g,-,. are coordinate-dependent when

we use spherical polar coordinates in flat Euclidean space. Thus we clearly have

to devise a means of extracting essential geometrical information distinctly from

pure coordinate effects.

1.1.3 Contraction and raising and lowering of indices

Contraction means, identifying a covariant index with a contravariant index of a

mixed tensor and this will reduce the rank of the resulting tensor by two [5, 6]. For

example, Alf, is a fourth rank tensor, where as is a second rank tensor, which is

evident from its transformation rule. The outer product of two tensors increases

the rank, ie., A’7B"’ = C‘-7"“ is a tensor of rank four. The inner product is defined

as an outer product followed by a contraction. For example, = {m is an

inner product and its rank is three (each contraction reduces the rank by 2).

Now consider associated tensors defined. by the following

A, = g,-kA'° A" = g“‘A,-, (1.9)

where g”° is the contravariant form of the metric tensor. These operations de­

fine the lowering and raising of indices by the metric tensor, respectively. The

significance of such rules is clear from the relation

g,-,,A*'A’= = .4,,A’°, (1.10)

which is a scalar. Thus from (1.8), it is clear that dsz is an invariant interval.

1.1.4 Covariant differentiation and parallel transport

We can readily show that, the ordinary derivative of a vector in an arbitrary

space-time does not transform like a vector [5, 8]. Consider the transformation

equation for a vector B,-:

I 6 iBk = 6713, (1.11)Here the prime corresponds to a different coordinate system. Differentiating w.r.t.

1"" we get

BB), 83:1 32:" BB, 822:1am ’ ark arm 3:5" + az'maz'kB‘ (L12)

Because of the second term on RHS, the ordinary vector derivative does not. . . 2 ' . .transform like a tensor. The second derivative ; is in general non-zero and

indicates that the transformation coefficient in eq. (1.11) varies with position in

spacetime. This property is not confined to non-Euclidean geometries. It also

holds in Euclidean geometry wherein non-Cartesian coordinate systems are used.

We have in general

331' Bi (£13k + 633") - Bi (ink)61:” = 6lim0 61” (1.13)

Here the difference in the numerator is not expected to be a vector, because the

two terms in the numerator do not transform like vectors at the same point, due

to the variation of the transformation coefficients with position. In order to find

the change in the vector between two points, we must somehow measure this

difference at the same point. This is achieved by a process known as parallel

transport, ie., shifting vectors from an initial to a final point without changing

its magnitude and direction. We can express the change in the vector 6B‘ due to

parallel transport as (keeping in mind a simple Euclidean example)

53,- = 1“§,B,6z‘= (1.14)where the coeflicients Fl,‘ are functions of space-time coordinates Ii The set

{P5,} constitutes the so-called affine connection on the space-time region. The

symbols are sometimes referred to as the Christoffel symbols (II kind). Now the

difference between the vector B, (wk + 62K) and the vector obtained by parallel

transport, B, + 6B, is a vector at an" + 6$'°:

8B,­

Bi(.’Ek + 63:") — + = — r§.,B,l 6a:'° (1.15)We may accordingly redefine the derivative of a vector by

B,-,,, = B,-_,, — r§,B, (1.16)This derivative must transform as a tensor. It is called a covariant derivative

(semicolon represents covariant derivative). Riemannian geometry introduces

further simplification[3]:

lcl = ll: §9ik;z = 0 (1-17)

Pi, and 91/: are related by

1 _ 9”" agmk 39m: 391::P“ ‘ 2 l 83:’ + ask azml (H8)

The Christoffel symbols ofI kind are defined by

. . 1 3g,~ 8g- 897;‘[1];]C]= E + — Tr)": (1.19)

Combining equations (1.18) and (1.19), we find

rig. = g"“° [z'j, m] (1.20)To this end, it may be noted that, by a linear transformation we can arrange to

have a coordinate system with

911 = 771:: = diag (1, -1, -1, -1), P1,: 0 (1.21)

at any chosen point in space-time. Such a coordinate system is called a locally

inertial coordinate system (or local inertial frame, LIF). The covariant derivative

reduces to ordinary derivative in a LIF. The significance of LIF becomes clear

when we discuss the Ei11stein’s equivalence principle [4, 5, 11].

1.1.5 Spacetime curvature and the Riemannian curva­ture tensor

Consider a triangle (say AABC) on the surface of a sphere (two dimensional).

Imagine a vector at the point A being parallel transported to C through two

different paths, ie., A to C and A to C’ through B. It is found that the final di­

rections of the vector are different in the two situations, ie., the outcome depends

on the path of transport from A to C However, if a similar experiment is

8

conducted with a triangle drawn on a flat piece of paper, there will be no change

in the directions of final vectors. This is one of the properties that distinguish

a curved space from a fiat one. The four dimensional curved spacetime may be

characterized by means of a fourth rank tensor.

Let us consider the difierence between two ‘2nd order covariant derivatives in

the form,

Bp;q;r ’ Bpmq = BmRZ:;r (1-22)

where

m 8 m 6 m ' m ' mRM, = fil"p, — axr PM + I‘;,,l“jq — Pgqfjr. (1.23)

Also

R,',’;, = —R;,’:§ (1.24)

From the quotient law we conclude that R,’,’;,. are components of a fourth rank

tensor. This tensor, known as the Riemann Christofiel curvature tensor, plays an

important role in specifying the geometrical properties of the four dimensional

spacetime Spacetime is said to be flat if its Riemann tensor vanishes every­

where. Otherwise it is said to be curved. Thus, in general, a curved spacetime

is characterised by coordinate dependent metric tensor components g,-k, (ii)

non-vanishing Christoffel symbols I‘;-k and (iii) non-vanishing Riemann curvature

tensor.

Properties of Riemann tensor

This tensor can be contracted (equating one covariant index to another con­

travariant index) in two different ways[4]:

Rgq, = o (1.25)and

Rggm = R,” (1.26)R1", is a second rank tensor called Ricci tensor. From eq. (1.23) the Ricci tensor

can be expressed in the following form

3 m 8 m ' m ' 171RP‘? = firpm _ W??? + rim 1'4 — Fin: jm (127)

Or equivalently

_ 8 8 BPS}? . m - <9log\/§B1.-,q — % lOg - arm + Pimp]-q —

where g is the determinant of the metric tensor. It readily follows from this

expression that the Ricci tensor is a symmetric tensor ie., R“ = Rqpu By further

contraction, we get a scalar called Ricci scalar, given by

R = g""R,pq (1.29)To see the other symmetry properties of the curvature tensor, we express the

Riemann tensor in the covariant form which is more convenient:

Rrpqr = Qmn R3} (1-30)10

The additional symmetries [4] are the following:

Rm, = —R,,,,q, (1.31)

Rm, = —R,,,,,q (1.32)

Rrlpqr = Rqrnp (1-33)and

Rm, + Rm. + R,.,,,., = 0 (1.34)In general a tensor of rank r in n-dimensional space has n’ components. However,

because of the symmetry relations, the Riemannian tensor has only 20 indepen­

dent components.

Bianchi identity

Rmpqrm ‘l’ Rrnprma + Rmpnmr = 0 (1-35)

The significance of the Bianchi identity is that it leads to a zero divergence

tensor called the Einstein tensor defined as

G'"‘’’ = Rm? — §g'"7’R. (1.35)

1.1.6 The Principle of Equivalence(PE)

The principle of equivalence [4, 5, 11] played a key role in GTR. The principle

of equivalence states that gravitational effects are identical in nature to those

arising through acceleration. The seed for this idea goes back to the observation

11

by Galileo that bodies fall at a rate independent of mass. In Newtonian terms,

the acceleration of a body in a gravitational field g is

mm. = mcg (1.37)and no experiment has ever been able to detect a difference between the inertial

and gravitational masses m I and ma (Inertial mass m; occurs in Newton’s second

law and gravitational mass mg occurs in Newton’s universal law of gravitation).

These considerations led Einstein to suggest that inertial and gravitational forces

were indeed one and the same. This leads to equivalence principle which is usually

stated in two different forms.

The weak principle of equivalence states that the effects of gravitation can

be transformed away locally and over small intervals of time by using suitably

accelerated reference frames. The strong principle of equivalence states that, any

physical interaction (other than gravitation) behaves in a locally inertial frame

(for example, freely falling lift) as if gravitation were absent. In other words, in

a small laboratory falling freely in a gravitational field, mechanical phenomena

are the same as those observed in a Newtonian inertial frame in the absence of a

gravitational field. In 1907, Einstein replaced the phrase ‘mechanical phenomena’

by the phrase ‘laws of physics’ and the resulting statement is the principle of

equivalence. These freely falling frames covering the neighbourhood of an event

are very important in relativity, they are called local inertial frames (LIFS). These

local inertial frames are characterized by

g,-k = 7},-k = diag (1, -1, -1, -1), P1,: 0

Thus in the LIF, gravitation has been transformed away momentarily and in a

12

small neighbourhood of any point in a curved spacetime. However, gravitation

cannot be removed globally and a curved spacetime is characterised by a non­

vanishing Riemann tensor.

1.1.7 Einstein’s field equations

Energy momentum tensor and the action principle

The energy momentum tensor plays a vital role in general relativity. In electro­

dynamics the electromagnetic energy momentum tensor T“‘ describes essentially

the conservation of energy through CT?" = 0. In general relativity T“‘ acts as a

source of Einstein’s field equations

The famous action principle was introduced in 1834 by Hamilton to obtain

the generalized laws of dynamics. The action [12] is defined through an integral

A = /:2 L(q,,q,,t)dt (1.33)1

The scalar function L is called Lagrangian, which is a function of generalized

coordinates qr, their time derivatives and time coordinate t. When the system

makes a transition from an initial state to a final state, the actual path is that

particular path for which A is stationary for small displacements of the path (ie.,

6A=0)

A more general form of action principle

Let a system be described by means of a set of functions d>A(A = 1,2,

of spacetime coordinates 2:‘ From d)‘ and its derivatives gbfi, construct a

Lagrangian density (scalar function)

L’ = L’ (¢", ¢>j3,z‘) (1.39)13

The action integral is defined as

A: /V L’./Tgafiz, (1.40)where V is the volume of the specified space-time manifold with the boundary

surface 2. The equations satisfied by <15‘ are such as to make 6A = O, for small

variations 6494- (43’‘ —> W‘ + 6¢"‘) which vanish on 2. The spacetime geometry is

specified by the metric tensor gik. If we demand that the guc are also dynamical

variables and that the action A remains stationary for small variations of the

type

gik 9 91'); + (sgik.Then the variation in action is expressed as

6A = —% [Tik 6g,-kx/—g d4:I: (1.42)

and this yields a definition of the energy-momentum tensor for the entire physical

system described by the action principle [4, 8]:

- _2 6 I 3L, —gwk = _ _L ,x-. _ (L) (1.43)\/‘9 [agik 39uc,z _,

The variations of A w.r.t. g,-k leads to energy-momentum tensor of various

interactions. The energy-momentum tensor of a fluid with density p and pressure

p in the generally covariant form is

T” = (p + P) u"u" - P 9"‘ (1-44)

where ui is the four-velocity of the fluid. It can be readily shown that

14

= 0 (1.45)This represents the conservation law for the energy momentum tensor T“°. Hilbert

derived the field equations of relativity from an action principle. The action is

given by 1 ,A = i /VR,/——g d“x + [VL ,/_—g d4; (1.46)

The variation of A with respect to g,-k leads to the following equation called the

field equation of GTR [3, 4, 8].

Rik - $9,-;,R = -AZT,-k (1.47)where the coupling constant K is given by (from Newtonian approximation) rs =

%9 = 877G (with c = 1). Before this derivation, Einstein, however formed his

equations of general relativity from some general considerations. According to

him, the energy tensor acts as the source of gravity. However, in order to get a

stationary universe, Einstein modified the first term of the action in (1.46) by

adding a constant term (cosmological constant). ie.,

_i_2/~: /v(1-2+ 2)\)\/——g d4~_r:—/VL’,/——g d“::: (1.48)

Then the modified field equation is

1

Rik — 59113 + A91"): = -f€Tik (1-49)When Einstein came to know about Hubble’s discovery of the expansion of the

universe, he abandoned the cosmological constant term in his field equation.

However, this A—term is one of the most intriguing factors in current theoretical

15

physics. In view of its application to cosmology, the A—term is usually taken to

the RHS of this equation, after making the substitution

A (1 50PA — 87rG ' )so that 1 .

Rik — §.9ikR = -P6[Tuc + PA Que] (1-01)

In the contravariant form we have

1' 1 i 1' iR’°—§g'°R=rc[T"+p,\g'°] (1.52)

From (1.44), we can see that the term p,\g"‘ is identical to the energy-momentum

tensor for a perfect fluid having density p,\ and pressure p,\ = —p,\.

1.1.8 Earlier solutions

The Schwarzschild solution

Karl Schwarzschild in 1916, solved the Einstein’s field equation to describe the

geometry of spacetime in the empty space outside a spherically symmetric distri­

bution of matter. The Schwarzschild metric is [1, 3]

v -1 ,d32 = (1— 20M) at? — (1 — 2G‘ 1) d7'2 — r2 [492 + sin2 9 d;- (1.53)7‘ T

Most of the traditional tests of GTR [8] are based on the Schwarzschild solu­

tion, and they seek to measure the fine differences between the predictions of

Newtonian gravitation and those of general relativity. The gravitational redshift,

precession of the perihelion of mercury, the bending of light, existence of black

16

holes etc. are some of the predictions of GT R. Among these, the precession of

the perihelion of mercury is perhaps the most impressive test in favour of GTR..

The theory predicts a precession of 43 seconds of are per century, and there is a

good agreement with observations. Eventhough the Schwarzschild solution rep­

resents the first physically significant solution of the field equations of relativity,

this is a local solution in the sense that the distortions of spacetime geometry

from Minkowski geometry gradually diminish to zero as we move away from the

gravitating mass. ie., spacetime is asymptotically flat and also static.

The Einstein solution

Einstein realized that the Schwarzschild solution cannot provide the correct space­

time geometry of the universe, since the universe is filled with a continuous dis­

tribution of matter.

In order to solve field equations of general relativity (which are an interlinked

set of nonlinear partial differential equations), it is essential to introduce certain

simplifying assumptions, just like the assumption of spherical symmetry in the

Schwarzschild solution. Einstein assumed homogeneity and isotropy in his cos­

mological problem. He further assumed that spacetime is static. Under these

assumptions, the line element of the spacetime could be described by [3]

(172

1 — 7'2ds2 = dt2 — a2{ + 7'2 ((192 + sin2 0 d<,02)}. (1.54)

The constant a is called radius of the universe. However Einstein failed to obtain

static, homogeneous, isotropic dense model of the universe from the field equa­

tions, using the above line element. Hence he introduced a cosmological constant

term in his field equation, which introduces a force of repulsion between objects,

to obtain a closed model.

17

Einstein believed that GTR can yield only matter filled spacetimes as so­

lutions of the field equations. However, it was proved wrong shortly after the

publication of his paper in 1917. W. de Sitter, a Dutch astronomer, published

another solution of the field equation which predicts an empty spacetime, but

expanding (ie., expansion without matter). It had the remarkable property of

predicting a redshift proportional to the distance. However, the de Sitter model

fails to meet Mach’s criterion that there should be a background of distant mat­

ter against which local motion can be measured. But the observations of Hubble

and Humason (1929) indicated that the universe is not static but expanding.

Einstein, then abandoned the cosmological constant term in his field equations,

remarking that it was the biggest blunder of his life.

The combined effect of deSitter’s notion of expansion and Einstein’s notion

of non—emptiness is obtained in the Friedmann model (1922). In 1922, the gen­

eral homogeneous and isotropic solution of the original Einstein equations was

found by the Russian mathematician Alexandre Friedmann. It is these Fried­

mann models based on the original Einstein field equations, and not the Einstein

or de Sitter models, that provide the mathematical background for most modern

cosmological theories. In 1929, Hubble’s observations regarding the redshifts of

galaxies established a linear relation between velocities and distances, indicating

that the universe is expanding, ie., galaxies are receding away from each other

(Hubble’s law)

1.2 The Standard Cosmological model

This model forms the most successful approximation of the real universe, with

maximum consistency with observations. The Friedmann models are the simplest

ones and are based on the following simplifying assumptions [1-4,13,14]:

18

1. The geometry of space is such that at large scales, it is describable by

the mathematically simple, spatially symmetric Friedmann-Robertson-Walker

(FRW) metric, which is based on the cosmological principle, ie., matter dis­

tribution in the universe is homogeneous (independent of location) and isotropic

(same in all directions) on very large scales.

2. The energy-momentum tensor is that which corresponds to a spatially homo­

geneous and isotropic perfect fluid comprising matter which is either relativistic

or non-relativistic.

3. The world lines of matter form a highly ordered non-intersecting bundle of

geodesics, which can be parameterized by three space like coordinates 1:“, ,u =

1, 2, 3. Thus 3“ =constant, along a world line. Also, there exists a set of space-like

hypersurfaces given by t =constant, orthogonal to this set of world lines. The

time t may called the cosmic time. The observers whose world lines follow the

above equation are called fundamental observers (Weyl postulate).

1.2.1 Robertson-Walker line element

The rigorous derivation of FRW metric, which is used by Friedmann and others,

had to await the work of H P Robertson in 1935 and A G Walker in 1936.

Independently, these authors showed that there are only three kinds of such

spacetime, denoted below by the parameter values k = 0, +1,& — 1. The line

element, whose derivation is given in many standard text books [4, 13], is given

2

(182 = dt2 ’ a2(t) + r2 [d02 + sin? 0 dcpzl} (1.55)

Here r, 9, «,9 are the three coordinates I“. The function a(t) sets the scale of the

spacelike sections spanned by r, 6, cp. a(t) is called the scale factor or expansion

19

factor for the universe. For k = 0, the expression in the bracket is simply the

line element for three dimensional Euclidean geometry, ie., for flat space. For

this reason this case is often referred to as the Hat Robertson-Walker model. For

= 1, the space is finite but unbounded with 0 3 1' 3 1 (space curvature is +ve).

This is the closed model, while for k = -1, the model is said to be open (spatial

curvature is —ve). Thus for k = 1, 0 or —-1, the three dimensional spatial part of

the metric is hyperspherical, hyperplanar or pseudo-hyperspherical, respectively.

For both k = 0 and k = -1, the coordinate 1" goes over the range 0 3 7‘ 3 oo.

The 9, go coordinates range over the intervals —7r 3 9 3 71' and 0 3 go 3 27r in all

three cases.

1.2.2 The Redshift

Consider a galaxy G1 at 7‘ = r1 emiting light and is received by the observer on

the galaxy G0 at r = 0. According to GTR, light travels along a null geodesic

ds = 0, which in this situation turnsout to be a path along which 9 and (,0 are

constants. From the FRW metric [3],

d7‘ —dt

where the minus sign on the RHS indicates that along the path of the light ray,

7' decreases as t increases. Let G1 emit monochromatic light of wavelength A1.

Consider two epochs at t1 and t1 + 6t1 when two successive wave crests leave G1,

reaching G0 at to and to + dto respectively. Then,

['1 dr [to dt [to+5to dt (1 57)0 V1 — /CT2 — ti a(t) — t1+6t1 a(t) IAssuming that the function a(t) varies only slowly so that it does not change

significantly over the small intervals 6150 and M1, we get

20

62:0 _ 6t;alto) 0 ($1)

Let the wavelength perceived by Go be A0. Then /\1 = c6t1 and A0 = cdto. Hence

(1.58)

:—:= =1+z (1.59)The parameter 2 is called redshift and a redshift of wavelengths indicates z > 0

and hence a (to) > a (t1) (provided to > t1). So, to be able to explain the observed

galactic redshift, it is necessary to have a (t) increasing with t.

1.2.3 Hubble’s law

The linear relationship between the distance to a galaxy and its observed red­

shift, may be deduced from the FRW metric, without specific knowledge of the

dynamics of the expansion The expansion of the universe means that the

proper physical distance between a pair of well separated galaxies is increasing

with time, ie., the galaxies are receding from each other. A gravitationally bound

system such as the local group is not expanding, and this gravitational instability

tends to collect galaxies into increasingly more massive systems that break away

from the general expansion to form a hierarchy of clusters.

Consider a galaxy having an absolute luminosity L and let the measured flux

be F It emits light at 7‘ = 7'1, 72 = t1 and we receive it at r = 0, t = to. Due to

the expansion of the universe,

L LF = = 1.6047m? (to) T? (1 + z)2 47713;? ( )

where

DL = no (to) (1 + z) (1.61)21

DL is called the luminosity distance of the galaxy and a(t0) is the scale factor for

the present universe. If 1' << 1, we can approximate the integral in eq. (1.57) to

write

By Taylor expansion near to we get

a (151) 2 a(to)+(t1—t0) (1 (to)

and hence

Using eq. (1.59) we have

(1+z)‘1=: :1—r1a(t0)—For z << 1 and for small T1,

and so

2 = HODL

ie., with Doppler velocity v = cz = z (c = 1)

’U = HQDL

22

(1:62)

(1.63)

(1.64)

(1.65)

(1.66)

(1.67)

(1.68)

where Ho = is the present value of Hubble constant. This is Hubble’svelocity-distance relation. The Hubble parameter H in general is a function of

time.

1.2.4 The Friedmann models

In the preceding sections, the dynamics of the expanding universe only appeared

implicitly in the time dependence of the scale factor a(t). To make this time

dependence explicit, one must solve for the evolution of the scale factor [14]

using the Einstein’s equations.

G; = R; — $5,112 = s7rGT,;', (1.69)where T; is the energy-momentum tensor (or stress-energy tensor) for the source

including matter, radiation, vacuum energy etc. The assumption of homogeneity

and isotropy of the cosmic fluid, implies-that T3‘ 5 (p = 1, 2, 3) must be zero, and

the spatial components T5’ must have a diagonal form with T11 = T22 = T It is

convenient to write [14]

T2 = diag ll’ (1?) , —p(t), -P(t), -p(t)l, (1-70)

where p (t) and p(t) are the energy density and pressure of the cosmic fluid (if we

treat it as an ideal fluid). From the Einstein’s equations, using the FRW metric,

we get the following equations which are known as the Friedmann equations of

cosmology:

(12 k 87rG'— — = — 1.71,1, + a2 3 p, ( )2d (12 k: + F + E = —87rGp (1.72)

These two equations together with the equation of state

20 = p(p) = wp (1-73)completely determine the three functions a(t),p(t) and p(t). In the dust ap­

proximation (as done by Friedmann), we take p = 0 (because the dust matter is

assumed to be collissionless) in the above equations.

From eq. (1.71), it follows that

(1.74)

and k a2 p _E - .17 ' 1] “-7”where we used the result that ('1,/a = H (t), called the Hubble parameter, which

measures the expansion rate of the universe.

a2 -0,?k — (12 E — 1] (1.76)where pc = 3H 2 / 87rG, called the critical density of the universe (critical because

it corresponds to the fiat case). If we use p/pc = Q, then

is «'12

For the present universe

is = H3 a3 (90 — 1) (1.78)Since Hgafi 2 0, there is a correspondence between the sign of k and the sign of

Q -1. ie., ' = +1 corresponds to 9 > 1 (p > pc), is = 0 corresponds to Q = 1

24

(p = pc) and k = —1 corresponds to Q < 1 (p < pc). Correspondingly, we have

closed, flat, and open models. Equations (1.71) and (1.72) can be combined into

a. single equation for ii:

ii 47rGE - ‘T (P+310) (1-79)and in terms of the Hubble parameter H (t), it leads to the Raychaudhuri equation

[13, 17]

2

= —H’ — $ (p + 3p) (1.30)I“- d (1 ii

H-5z(;l—;'aFor matter (non-relativistic), p + 319 > 0, implying that EL < 0. The a(t) curve

N)

(which has positive cl at the present epoch to) must be convex. ie., a will have

been smaller in the past and becomes zero at some time in the past (say t = 0).

The time span to must be less than 1/Ho. As a —> 0, p —> oo and the components

of the curvature tensor diverge. This is the singularity problem of the standard

model, which is an artefact of the theory. Also, when the radius of curvature

of spacetime becomes comparable to the Planck length (lp = % 2 10‘33cm),

quantum effects of gravity [see 90] become significant, and classical Einstein’s

GTR becomes invalid. For a resolution of this problem, a quantum theory of

gravity is needed. Equations (1.71) and(1.72) can be combined to yield another

equation

gt; (a2+k)] = a [2aéi+c'z"'+k] (1.81)Of

d 3 2E(pa)+3pa :0 (1.82)25

This result is a consequence of the conservation law implicit in Einstein equations

T,§;,- = 0 (1.83)Eq. (1.82) is called the conservation law, because it corresponds to the conserva­

tion of the energy-momentum tensor of the universe. For an equation of state of

the form by eq. (1.73),

p oc a‘3(“'"’) (1.84)and from conservation law,

p = —3H (p + p) (1.85)For non-relativistic matter (11; = 0) => p oc (173, for radiation (w = 1/3) => p or

a‘4 and ifp = —p (for example, vacuum) 11) = -1 and p = constant. Ifp = —p,

pressure is negative (since p > 0, to maintain > O) and the negative pressure

allows for the energy inside the volume to increase even when the volume expands

(the case of inflation). From Friedmann equations

d2 —3(1+ w)E ot a , (1.86)(1 oc a-%<1+3'"> (1.87)

Integrating

a(t) oc t2/(3(1+"’)) w # -1, (1.88)

a (t) oc e“, w = -1 (1.89)26

where A is some constant. For w = 0, a or t2/3, for w = 1/3, a or W2 and for

w = 1, a 0: t1/3 If we assume matter and radiation are the main components of

p, and each is conserved separately (the assumption in the standard model), then

the present universe is matter dominated. There is an epoch (recombination or

decoupling epoch, td) at which p, = pm, and before that the universe is radiation

dominated. At td, matter gets decoupled from the radiation background, and

the universe becomes transparent. There is also another parameter used in FRW

models, called the deceleration parameter q (t), given by

Z = —qH2 (1.90)A +ve value for q indicates decelerating expansion of the universe. If the present

value of the deceleration parameter is qo = 1/2, it corresponds to a flat universe

(I: = 0). qo > 1/2 leads to closed universe and qo < 1/2 corresponds to open

model.

1.3 Structure formation

One of the outstanding problems in cosmology today, is undoubtedly the origin

and evolution of large scale structures [13, 15, 16, 17]. The basic framework for

structure formation requires that small density perturbations, formed in an oth­

erwise uniform distribution of matter and radiation in the very early universe,

grow under the influence of gravity until gravitational instabilities develop and

the structure collapses and galaxies, clusters of galaxies etc. that we see today are

formed. The formation of structure (or galaxy formation) began when the uni­

verse became matter dominated, or we can say that the time of matter-radiation

equality is the initial epoch for structure formation. Thus the structures we see

today, are formed by a process known as gravitational instability, from primor­

27

dial fluctuations in the cosmic fluid (which is evident from the small anisotropy

in CMBR). But, because the strength of clustering is expected to increase with

time (ie., the evolution of the density contrast is proportional to some power

of scale factor in the linear approximation of the standard model), the galaxies

must deviate from the smooth Hubble expansion. These deviations away from

uniform Hubble flow are known as peculiar velocities. According to standard

model, 612 oz Q8'56p, where 6p is the density perturbation, and Q0 is the present

value of the ratio between critical density and density of the universe.

1.3.1 Linear perturbation theory

We assume that at some time in the past, there were small deviations from

homogeneity in the universe. As long as these inhomogeneities are small, their

growth can be studied by the linear perturbation theory. On each hypersurface

(:z:“ = constant, p = 1, 2, 3), one can define an average plus a perturbation [17]

MI, 15) = pa (t) + 5p (N) 7 (1-91)

12 (I, t) = pa (t) + 51> (I, t), (1-9?)

H (:5, t) = H,, (t) + 6H (:5, t) (1.93)

Here If is the time coordinate labelling the hypersurfaces, and 2: = ($1, :52, :3) are

space coordinates. Density contrast is defined by

6 ,t6 (:5, t) = p(-7: )

Pb

In the first order linear perturbation approximation, the conservation equation

(1.94)

(1.85) remains the same, but the Raychaudhuri equation becomes, to first order

28

- 47rG 1 V26p__ 2____ __H— H (p+3p) 3(p+p) (1.95)

V2 is the Laplacian on a comoving hypersurface, given in terms of comoving

coordinates by

V2 = a‘26""6,-6,The energy conservation equation (1.85) and Raychaudhuri equation (1.95) de­

termine the evolution of the energy density and the Hubble parameter along

each world line, including first order perturbations away from homogeneity and

isotropy. The perturbation equations [15, 16] obtained are the following

(5?) = ‘3 (Pb + 3171:) 5H — 3Hb‘5P1 (1-97)

477G Vzclp5H =—2HaH——5 ———, 1.93( ) b 3 p (Pb+Pb) ( )where an overdot denotes differentiation with respect to time. In the linear

approximation (ie., 6 < 1), when perturbation from the background density is

small, the evolution of density contrast is given by

626 (106 VzpW-F235;: W+47rG'pg,5 (1.99)

For dust models, one can solve the above equation to yield 6 proportional to some

power of the scale factor a, ie., 6 oc a in the matter dominated case and 6 oc a2

for radiation dominated epoch. When we take two or more component cosmic

fluid, instead of treating them separately, one can still find solutions for 6, but

the process gets highly complicated.

The linear perturbation theory fails when the density contrast becomes nearly

unity. Since most of the observed structures in the universe, for example, galaxies,

clusters etc. have density contrasts far in excess of unity, their structure can be

understood only by a fully nonlinear theory [15-19]. Although local extensions of

linear theory do provide a qualitative first step in comparing theory with observa­

tions, a deeper insight into gravitational clustering is provided by the dynamical

approximations, like the Zeldovich approximation, adhesion approximation etc.

Several approximations have been suggested to model gravitational instability for

the strongly non—linear regime (6 Z 1).

1.4 Predictions and problems of the standardmodel

The Friedmann models are the most successful approximation of the real universe,

with maximum consistency with observations. Now we consider the validity and

the problems of this theory through cosmological observations.

1.4.1 Hubble’s expansion and the redshift-apparent mag­nitude (z — m) relation

Modern observational cosmology began with Hubble’s observations. He obtained

a value for the Hubble constant, Ho = 500km s“Mpc‘1, from his observations.

However, present day observations suggest that Ho lies in the range 50 5 Ho 5

100 km s"1Mpc‘1[21, 22]. It is usually expressed in the following form, Ho =

100h0 km s‘1Mpc‘1, with 0.5 3 ho 3 1 being the uncertainty in the measured

value of Ho. The Hubble constant relates the redshift z of a nearby galaxy to

its distance D by v = cz = HOD. Thus if we measure 2 and D for a number

of galaxies, we should be able to estimate Ho. However there is difficulty in

30

estimating D. How is it possible to measure the distance D? All the distance

measurements are based on the assumption that recognizable types of distant

objects are similar to nearby objects of the same type. Let L and l be the

absolute and apparent luminosity of an object, respectively, then [3, 7]

L

l 2 47rDi(1.100)

where D; is the luminosity distance of the galaxy given by DL = T]-a (to) (1 + z),

with r,- the radial coordinate of the galaxy emitting light at some time tj in the

past and 2 its re_dshift. a (to) is the present value of the scale factor (Our galaxy

is at r = O and receiving light at to). In fiat F RW models

r,- = % (1.101)From (1.88) it can be shown that

1:3/3 to dt 2 1-=— —=—1— 1.102T] a(to) t,- 152/3 Ho 1+2 ( )so that

2

DL= — [(1+z) —\/1+2] (1.103)H0

A more general form of luminosity distance [3] is

_ 1 1/2D), _ Hoqg {qoz + (qo 1) [(1+ 2zqo) 1]}, (1.104)

where qg is the present deceleration parameter. lqo — > 0 for k = :l:1 models

and qo —> 1/2 leads to the flat case (1.103). Astronomers, instead of using the

power 1 as a measure of apparent brightness, use a logarithmic measure, the

apparent magnitude m. This is greater, the fainter the object, and is defined so

31

that two objects whose luminosities (Z1 and lg) differ by a factor of 100 differ in

apparent magnitude by 5, that is

llE = 100("'2-"*0/5 (1105)

Hence

Tng — 7711 = lOg1oThe absolute magnitude M of an object is the apparent magnitude that the

object would have at a distance 10pc Thus measuring D in pc (pc or parsec

means parallax second. One pc is the distance to an object whose parallax is

one second of arc with respect to a baseline which is usually the diameter of the

earth’s orbit around the sun for astronomical purposes. 1pc = 3.26 light years

= 3.08 x 1O13cm)

M-m10 5 = —— .0 D2 (1 107)and

m—M=5logDpc-5 (1.108)However cosmologists measure distance in megaparsecs (1Mpc = 105pc)

m—M=5logDM,,c+25=,u (1.109)

p is called the distance modulus of the object. Substituting for DL, we get the

redshift-apparent magnitude relation. Using Ho = 100hg ls:ms‘1Mpc‘1 one can

arrive at a useful formula [3]

32

1 — 1 1#0 = 5 log [h—o (g + (qoqg ) [(1 + 2zq0)E — 1]) }—2.5log (1 + z)+K (z)+42.39

(1.110)

where K (2) is called the K —correction term, which allows us to obtain the rele­

vant absolute magnitude corresponding to zero redshift [20]. Observationally H0

is measured from the z — m diagram. The FRW models predict a linear redshift­

magnitude relation. However observations of distant extragalactic objects like

supernovae, quasars etc. indicate that the diagram is a scatter diagram, which

increases with z, with no Hubble type relation clearly apparent [20, 23-28]. The

Hubble’s law is the foundation on which the expanding universe models rest. If

the law is known to be valid for all extragalactic objects, then only can we use it

to claim that an object at high z is farther away from us and is being viewed at

an earlier epoch than an object of low redshift. This problem and a new model,

which provide an alternative explanation for the puzzle are discussed in detail in

Chapter 2 of this thesis.

1.4.2 Fractal distribution of galaxies

The basic assumption of standard cosmological model is the Einstein’s cosmo­

logical principle which, in fact, is the hypothesis that the universe is spatially

homogeneous and isotropic on large scales. The earlier large-scale surveys of

galaxy distribution are based on visual counts of galaxy angular positions. Such

surveys extend to depths nearly 10% of the present Hubble radius. Redshifts are

much more powerful tracers of structures than angular positions alone, because

the redshifts reduce the ambiguity in distance. These surveys show clumpy small­

scale distribution. One measure of this is the two-point correlation function f(r)

[15], defined by the joint probability of finding a galaxy in each of the volume

33

elements dV1 and dlé at a separation r,

dP = n2dV1dVg[1 + §(r)] (1.111)

where n is the mean number density of galaxies. It is found that [1, 15, 16], for

r < 10 h‘1Mpc, §(r) or 7"”. Such correlation functions indicate that the small­

scale galaxy distribution approximates a nested clustering hierarchy, or fractal

with dimension D1 = 1.2. In the following subsections, we will discuss briefly the

topics, fractals and fractal dimension.

Fractals

Fractals are geometrical objects [29-32] that are self-similar (or scale—invariant)

under a change of scale, for example, magnification. This means that, if we cut

out a portion and then we blow this piece up, the resulting object will look the

same as the original one. It was B B Mandelbrot, who coined the term, fractals,

for those complex structures to express that they can be characterized by a non­

integer (fractal) dimensionality. Although Euclidean geometry and the theory of

smooth functions can describe regular shapes and forms (e.g., lines, planes and

differentiable functions), the concept of fractal geometry is needed for describ­

ing irregular shapes and forms as well as the behaviour of extremely irregular

mathematical functions. The branching of trees and their roots, blood vessels,

nerves in the human body etc. have fractal properties. Other examples include

a landscape with peaks and valleys of all sizes, a coastline with its multitude of

inlets and peninsulas, the mass distribution within a galaxy, the distribution of

galaxies and clusters in the universe and so forth.

Fractals are either mathematical or natural ones. The Koch curve [32] is a

good example of a mathematical fractal, which can be used to mimic a coastline.

34

The shapes and patterns found in nature are usually random fractals (for e.g.

the galactic distribution). The reason is that, they consist of random shapes

or patterns that are formed stochastically at any length scale. Because of the

randomness, the self-similarity of natural fractals is only statistical, ie., given a

sufficiently large number of samples, a suitable magnification of a part of one

sample can be matched closely with some members of the ensemble of samples.

Fractal dimension

A correct definition of a fractal set is a ‘mathematical object’ whose fractal or

Hausdorff dimension (D1) is strictly larger than its topological dimension and

less than the dimension of the embedding Euclidean space [29, 31]. For example,

in the case of a straight line, a magnification by a factor 3, increases its length

by 3 = 31. For a square, when the side is magnified by 3, its area is magnified by

9 = 32. For a cube the volume increases by 33 In general the magnification is

3D 1, where D1 is the dimension which is integer for all these three cases. However,

for a Koch curve, it is found that the magnification is by 3D1 = 4, which gives

a fractal dimension D1 = ln 4/ ln3 = 1.26. The fractal dimension of a typical

coastline is 1.2, which is different from the dimension of the embedding space,

but is close to that of a Koch curve.

Measuring the volume of a fractal embedded in a d—dimensional Euclidean

space, leads to the conclusion that they are objects having no integer dimension.

To determine the volume V of a fractal structure of linear size L, the structure

is covered by N(L) number of boxes of unit volume, hence V(L) = N(L). For a

fractal, N (L) diverges as L —> oo, according to a non-integer exponent,

N(L) oc LD’ (1.112)35

hence the dimension is

ND1: lim 1" (L).Lam IHL (1.113)

Now consider a fractal of finite size. Let N(l) be the number of d—dimensional

boxes of side I needed to cover the structure. Then N(l) diverges as Z —) 0

according to

N(l) o< 1-01 (1.114)Therefore

(1.115)

For non-fractals D1 = d.

Since fluctuations are always present in physical processes, they never lead

to structures with perfect symmetry. For instance, the random walk of particles,

diffusion limited aggregation of particles etc. lead to fractal geometry. In the

case of natural fractals it is more effective to calculate the so called density­

density correlation function [31] (since natural fractals are scale-invariant only in

a statistical sense),

f(1")=—‘17Z,p(T+'rI)p(r’), (1.116)which gives the probability of finding a particle at 7' + 1", provided there is one

at r’ An object is non-trivially scale invariant, if its correlation function is

unchanged upto a constant under rescaling of lengths by an arbitrary factor q:

E(qr) 0< q‘°E(T), (1-117)36

where 02 is some non-integer number greater than zero and less than d. It can

be shown that the only function which satisfies this equation is the power law

dependence of f(r) on r ie.,

f(r) oc F“ (1.118)Hence

N(L) oc f0LE(r)ddr oc Ld“’ (1.119)Using (1.112), we have D1 = d— oz which is the fractal dimension. The two point

correlation function of galactic distribution leads to a fractal dimension of ~ 1.8

[15, 16] for r 5 10 h‘1Mpc. However, many recent redshift surveys have revealed

that, the three dimensional distribution of galaxies and clusters are characterised

by fractal behaviour [33-42]. This has confirmed the de-Vaucouleurs power-law

density-distance relation, p (r) oc (r/ro)D‘_3, with a fractal dimension D1 z 2 at

least in the range of scales 1 to 200 h.‘1Mpc. This fractal behaviour of galaxy

distribution within a scale of ~ 2OOh‘1Mpc (this scale may even deeper and the

switch over scale to homogeneity is not yet identified) is a challenge to standard

cosmology. Also, for a homogeneous distribution of galaxies, Hubble’s count law

is [1]

N (< m) o(10°'2D"" (1.120)

N is the number of galaxies brighter than the magnitude m. D1 = 3 corresponds

to the standard case. Observations show relative, persistent fluctuations in the

number count versus m relation, which cannot be accounted for by the homoge­

neous distribution of the standard model. This problem is discussed in the final

37

chapter of the thesis.

1.4.3 The abundance of light nuclei

One of the fundamental problems of cosmology is to explain the primary creation

of matter in the universe. It is generally understood that nuclei with atomic

number A 2 12 are synthesized in stars through nuclear reactions. The nuclei

Li5, Be9, B10 and B” could be produced in galactic cosmic rays by the breakup

of heavy nuclei as they travel through the interstellar medium. It is believed that

the observed abundances of deuterium(D) and helium(He) is through the process

of nucleosynthesis [3, 13, 43, 44] of elementary particles, beginning with that of

the neutron and proton. The pioneering work in this field was done by George

Gamow in the mid 1940s. Gamow was concerned with the problem of the origin

of elements. He described the formation of nuclei of He and D starting from

protons and neutrons by nuleosynthesis in the early universe, when temperature

of the universe was of the order of 109K. However, by that time, Burbidge et al.

[45] demonstrated that such nucleosynthesis can take place in stars also. Instead,

if stars are able to achieve the objective of explaining the abundances of all

elements observed, we need not consider this a cosmological problem. However,

there is some doubt whether stars can do this entirely on their own, and that

is why cosmology becomes important. The doubt centers around the relative

abundance of He, D and H. The ratios are approximately of the order He/H 2

0.3 and D/ H '2 few times 10‘5 The ratios are by mass densities; the stellar

nucleosynthesis is unable to explain this. It is found that for our galaxy, unless the

stars were much brighter in the past, the process can account for only at most 30

percent of the above value. In standard cosmology, Gamow successfully explained

the nucleosynthesis of He and D (not heavier elements), in the first second or so

38

after the big bang. The high temperature of the radiation dominated early phase

is just right for this process.

1.4.4 The microwave background

The cosmic microwave background radiation (CMBR) provides the most funda­

mental evidence that the universe began from a hot early phase [1, 3, 13] (big

bang cosmology). Gamow and his collegues Alpher and Herman predicted that

the photons of the early hot era would have cooled down to provide a thermal

radiation background in the microwave region of the spectrum at present. In 1965

Arno A Penzias and Robert W Wilson of Bell Telephone Laboratory at Holmdel,

New Jersey, detected the CMBR, with a black body spectrum.

In the subsequent phases after nucleosynthesis, the universe cooled as it ex­

panded, and the temperature falls as [3]

1T —. 1.121cc ,1 < )However in those phases, electrons act as scattering centres for radiation and

the universe was quite opaque. As temperature lowered, H atoms are formed and

electrons are slowly removed from the cosmological brew, and as a result, the main

agent responsible for the scattering of radiation disappears from the scene. The

universe becomes transparent and this epoch is called the recombination epoch.

This also corresponds to the decoupling epoch, since matter gets decoupled from

radiation (this corresponds to 2 ~ 103). The radiation temperature falls from this

as the universe expanded, and at present its value is T, = 2.736 :l: 0.017. This

CMBR across the sky is highly isotropic and uniform. The temperature of CMBR

across the sky, is reasonably uniform with % 2 1O"5 on angular scales ranging

from 10 are seconds to 180° "[13]. The primeval density inhomogeneities necessary

39

to initiate structure formation result in predictable temperature fluctuations in

the CMBR, and so the anisotropies of CMBR provide a powerful test of theories

of structure formation. The discovery of CMBR 35 years ago, had a profound

effect on the direction and pace of research in physical cosmology.

1.4.5 Other problems with the standard model and theinflationary model

Apart from the problems we have already pointed out, there are other serious

problems with the standard picture. Eventhough, the discovery of CMBR made

a widespread acceptance of the standard model, there are several puzzles to be

solved, with the standard picture. The most outstanding among them are the

following.

Singularity problem

From equations (1.84) and (1.88), it is evident that, in the standard model, the

scale factor vanishes at some time t = 0, and the matter density at that time

becomes infinite (also the temperature). It can be shown that at that time, the

curvature tensor Rijkl goes to infinity, ie., geometry itself breaks down at that

instant. This is unavoidable in the theory. This point t = 0 is known as the cos­

mological singularity or big bang. ie., spacetime was singular at that epoch. One

of the most puzzling questions facing cosmologists is whether anything existed

before t = 0. The universe came into existence at this instant, violating the law

of conservation of energy. This is the singularity problem [46, 47].

Flatness problem

From the Friedmann model we have (see eq. (1.77),

40

19 — 1| = lililikl = [a(t)]" (1.122)C

pc is the critical density for flat universe. The present value of Q is not known

exactly (0.1 3 Q0 3 2). But (12 oc 1/1‘. in the early evolution of the universe,

l9 - 1| = f; — 1| was extremely small. For the present 90 to be in the given

range, 9 at early times was equal to 1, to a very high precision (extremely high

fine tuning). It can be shown that

§2(1o—“3s) = 1 i 0(10'~">7). (1.123)

§2(1s) = 1: O(10‘“‘). (1.124)

This means that if Q at Planck time (t,, = 5.4x 10‘44s) was slightly greater than 1,

say Q(10"43s) = 1+1O’55, the universe would be closed, and would have collapsed

millions of years ago. If Q is slightly less than 1, ie., Q(10"’3s) = 1 — 10'“, the

present energy density in the universe would be negligibly small and the life would

not exist. In the standard model, it is not clear why the universe was created

almost flat, with such accuracy (fine tuning). This is the flatness problem [3, 46].

Horizon problem

An observer at r = 0 at a given time t can communicate to the maximum distance

2ct in a time interval 15 (c is the speed of light, which is the limiting speed). This

represents the radius of the observer’s particle horizon. For the GUT era (where

unification of three basic forces, namely strong, electromagnetic and weak, take

place), the temperature is T ~ 1015Gev and t = 10‘35s after the big bang, the

particle horizon is of the order of 6 X 10‘25cm. Suppose the universe expanded

41

as in Friedmann models till the temperature has dropped from 1015Gev to 3K ~

3 x 10‘4ev (present epoch). From thermodynamical considerations we can show

that the temperature varies inversely as the scale factor for the universe(a(t)).

Therefore the scale factor increased by a factor 3 X 1027, and the particle horizon

becomes 180cm only. Since no physical interaction travels faster than light, the

particle horizon sets limits on the range of causal influences. Therefore one can

not expect homogeneity to be established beyond this range. Then how come

the universe is homogeneous on large scales observed today. This is called the

horizon problem [3, 46].

Problem of small scale inhomogeneity

The universe is not exactly homogeneous (because hierarchical structures are

present). However, on very large scales~ 4000 h‘1Mpc, it is believed to be

homogeneous and isotropic. In the early epochs, before recombination (matter­

radiation equality) the universe is assumed to be homogeneous and isotropic,

which is quite reasonable, due to the remarkable discovery that CMBR has uni­

form temperature on all angular scales. However, recent measurements show

anisotropies in CMBR (% ~ 10-5) and this indicates inhomogeneities in the

matter distribution of the universe at the time of decoupling (recombination).

After decoupling, these irregularities grow under gravitational instability. The

density contrast is usually expressed [13] in a Fourier expansion

pg, is the background density. is is the comoving wavenumber associated with a

given mode, and 6,, is its amplitude. So long as ff << 1 (linear regime), its

physical wavenumber and wavelength scale simply with a(t): kphy, = k/a(t),

42

Aphy, = a(t)2T". Once :5)‘: becomes 2 1 (nonlinear), it separates from the general

expansion and maintains an approximately constant physical size (For example,

on the scale of galaxy 6p/pb ~ 105). And also from the fact that in the linear

regime, 6p/pg, grows as a(t) during matter dominated epoch, we can infer that

perturbations of amplitude 10"5 or so must have existed on these scales at the

epoch of decoupling. It should be possible to account for the anisotropies in

the CMBR (observed by COBE satellite) on this basis. However, the problem

in the standard model, is that at the time of decoupling (z ~ 103), the Hubble

radius (c/ H ) subtends an angle of only 0.8° on the sky today, while CMBR shows

anisotropies on all angular scales. The difficulty is that, if one imagines causal,

microphysical processes acting during the earliest moments of the universe and

giving rise to primordial density fluctuations, the existence of particle horizons

in the standard model precludes production of inhomogeneities on the scales of

interest [13].

Entropy problem

The total entropy (S) of the observable part of the universe is of the order

(aoT.,)3 ~ 1037, where an is the present scale factor and T., = 2.7K is the temper­

ature of CMBR. The entropy at Planck epoch will also be of the same order in

the standard picture. In a different way, we can say that the ratio of the number

of photons to baryons is 103 to 101° in the universe. Why is this number so large?

This is referred to as the entropy problem [46, 47].

The monopole problem

This is related to the horizon problem. The grand unified theories predict that

as the universe cools down and the temperature reaches ~ 1023K, a spontaneous

43

symmetry breaking occurs and as a result, magnetic monopoles are produced.

However no such monopoles have yet been detected. This is the monopole

problem [3, 46]. Maxwell’s electrodynamics does not permit the existence of

monopoles, but they do arise in the above scenario. The mass of the monopoles1015Gev

C2so produced is ~ ~ 10‘3g. If one monopole is created in the horizon size

region, ie., in a spherical volume of radius 6 x 10‘25cm, its density is

10‘3gPmonopale = . (1.126)3

The present density in Friedmann model is

_ 10‘3g’°”‘°"°”°“ ' (“3—"')(6 x 10-26)3(3 x 1027)3cm3

2 3 x 10-15;, cm"‘ (1.127)

This value of the monopole density is much larger than the cosmological density

(10‘29g cm‘3) of the present universe, and if monopoles existed at this rate, the

universe would have collapsed much earlier.

Cosmological constant problem

The cosmological constant, first introduced by Einstein as an arbitrary parame­

ter, is seen in contemporary physics to have a quantum origin. It is related to

the vacuum energy of the universe, with an equation of state p,, = —p,,. From

the cosmological observations, it follows that the vacuum energy density in the

present universe can not be much greater than the critical density of the universe.

ie., p,, 3 10‘29g cm"3 If p,, is viewed as arising from the potential energy of a

scalar field (V(q‘>)), then quantum field theory can not account for this vanishingly

small value for p,, compared with Planck density pp; = 0.5 x 10949 cm'3 This is

44

one of the most difficult problems of unified theories with spontaneous symmetry

breaking. This is called cosmological constant problem [46, 48].

Inflation

Many new models [49-56] were proposed during 1980s to overcome some of these

problems with the standard picture. In 1981, Alan Guth proposed [57, 58] and

later improved [59—63] by Linde and Albrecht and Steinhardt, the so called in­

flationary model as the solution to these problems. Inflation means rapid or

exponential expansion ie., a (t) 0: e“ where /\ is a constant. According to the

inflationary universe [46, 57-65] scenario, the universe in the very early stages of

its evolution was exponentially expanding in the unstable vacuum-like state (of

quantum mechanical nature). At the end of the exponential expansion (inflation)

the energy of the unstable vacuum (of a classical scalar field or inflation field)

transforms into the energy of hot dense matter and the subsequent evolution of

the universe is described by the usual hot big bang theory (or standard model).

The inflationary universe scenario makes it possible to obtain a simple solution

to many long-standing cosmological problems and leads to a crucial modification

of the standard picture of the large-scale structure of the universe.

The basic idea is that, there was an epoch when vacuum energy (12,, = —p,,)

was the dominant component of the energy density of the universe, so that scale

factor grew exponentially (see eq. (1.89)), ie.,

a(t) 0: exp < §2—GV(¢) t) (1.128)where V (<15) is the potential energy of the scalar field or inflation field. During

such an epoch, a small, smooth and causally coherent patch can grow to such

a size that it easily encompasses the entire observable universe today. The size,

45

during a small interval of time rises by a factor of ~ 105° This inflationary

scenario solves flatness, horizon, entropy and monopole problems of standard

model

However, there are still problems with this inflationary models. One of them

is the age problem. According to inflationary models, the present value of the

density parameter Q0 lies close to unity. This in turn leads to Hoto ~ 2/3.

However, present estimates [21, 25, 70, 74] put this value in the range 0.85 <

Hoto < 1.91 contrary to the above prediction. This is called the age problem in

the standard and inflationary models.

If we postulate a non-zero relic cosmological constant in the present universe

with density (p,\) comparable to matter density (pm) to overcome the age crisis,

then the problem is that this cosmological constant is indistinguishable from

the vacuum energy which produces inflation, and the model is bound to explain

how this vacuum energy does manage to change from large initial magnitude in

the early universe to a very small value at present. This will require extreme

fine tuning, which is not different from the fine tuning problems in the standard

model [3, 46].

Inflation does not solve the small-scale inhomogeneity problem. It also shed

no light on the singularity problem of the big bang cosmology, because the infla­

tionary stage is expected to occur at a time many orders of magnitude greater

than the Planck time. In order to overcome these problems several new cosmo­

logical models were proposed [49-56] during the eighties and nineties.

46

Chapter 2

The New approach

2.1 Introduction

The majority of cosmologists have by now taken for granted that the standard

hot big bang model of the universe is the correct starting point for the study of

cosmology. The hot big bang model, which is the most successful approximation

of the real universe, is based on the assumption that the matter distribution

in the universe is homogeneous and isotropic on the large scale average. This

interpretation leads to the prediction of Hubble’s law that the apparent recession

velocity of a galaxy is proportional to its distance, the constant of proportionality

being the Hubble parameter H Observationally, the Hubble parameter is

found from the redshift-apparent magnitude (2 — m) diagram or Hubble diagram

(Fig. 1).

The uncertainty in the determination of the Hubble parameter [24, 25, 26,

27, 66], which is a measure of the expansion rate of the universe, is one of the

most intriguing issues in the history of cosmology. The origin of the uncertainty

is obvious from the Hubble diagram; despite rigorous attempts to control random

errors and systematic effects (for example, the effect of dust grain in the region

between stars and galaxies, effect of metallicity or chemical composition etc.

47

which are common to all types of measurements) in measurements [66], there

is a clear scatter in it, which is evident in the latest Type Ia supernovae data

(from supernovae cosmology project, in papers by Perlmutter et al. [25] and

Riess et al. ]26]) with no deterministic Hubble type relation clearly apparent.

The same is true for a collection of high redshift quasars also [20, 23, 24], with

1 < z < 5 (see Fig. 1), where the points are widely scattered‘. However, the

distance measurements are extremely difficult for quasars, due to the difiiculty

in identifying the standard candles, the scatter may be caused by the variation

of intrinsic luminosities of quasars of the same 2. Once this is admitted, the

cosmologists can no longer claim that a faint object is necessarily far away. Some

authors have proposed non-cosmological contributions to the redshift as a possible

explanation for the scatter [20, 24, 28]. The vertical scatter in the Hubble diagram

could be due to objects of varying redshifts with the same luminosity. This

possibility admits an intrinsic non-cosmological component in the redshift 2, like

(1 + z) = (1 + z,-) (1 + 26) with cosmological component 2,, obeying Hubble’s law.

The intrinsic component 2, may have origin in Doppler and gravitational effects

or due to some difference in the age of the objects. The data is more accurate

for supernova (SNa), and the scatter in its diagram needs explanation. It is also

found that the scatter increases with redshift. The aim of this work is to offer an

alternative explanation for the above puzzle.

Variations in the expansion rate due to peculiar velocities are a cause of

error in measuring the true value of Ho (present Hubble parameter) [66]. These

are supposed to be induced by observed density fiuctuations. Since the density

fluctuations are evolving phenomena, peculiar velocities induced by them cannot

lead to the large randomness observed at early epochs, though it is a feasible

1The figure is taken from Ref [20], page 407

48

24 | «IIIIII I I‘| lTx|I' T 1lT‘In.

l\) l\) [*1

I__I_l_I__1_I_l‘__1_1_1_‘_

18­

Apparent magnitude

I ' v v I ll’ ' ' 1 ' I ' | 1*'v'01 0.1 1 10Log redshift

The reds'nifi—magn1Iude plot for quasars from Ih: Hewitt and Burb:cL~_:: catalogue.

physical process in the late universe.

In the stochastic model herein reported, we have attempted to explain this

scatter as arising from an inherent stochastic or non-deterministic nature of the

Hubble parameter, which provides a possible potential source of uncertainty in the

measured value of H0. In the following section we present the assumptions that

lead to a stochastic nature for the Hubble parameter and derive the probability

distribution function (PDF) for H using the Fokker-Planck formalism.

2.2 Stochastic equation of state

Since high values of 2 means we are probing into early epochs, the scatter di­

agram mentioned above indicates that the behaviour of the Hubble parameter

is anomalous in the early universe. The scatter indicates the uncertainty in the

measured value of Ho. Here we formulate an alternative scenario in which the

Hubble parameter or the expansion rate of the universe is a stochastic variable,

where the dynamics of the universe is described by non-deterministic Langevin

type equations.

It may be noted that in the standard cosmology, the cosmological fluid is not

uni-component. In fact, it is known that matter and radiation (with equations

of state pm = 0 and p, = %p,, respectively) in disequilibrium coexist in many

elementary subvolumes of the universe [67]. Some recent measurements [25, 26,

48, 68-80] on the age of the universe, Hubble parameter‘, deceleration parameter,

gravitational lensing etc. point to the need of extending the standard model

by including more components other than matter and radiation to the energy

momentum tensor of the present universe. Recent observations suggest that a

large fraction of the energy density (p) of the universe has negative pressure. One

explanation is in terms of vacuum energy or cosmological constant (with equation

49

of state p,, = —p,,) while another is in terms of quintessence in the form of a

scalar field slowly evolving down a potential (with equation of state pq = wqpq,

-1 < wq < O). The gravitational efiects of such components are opposite to that

of non-relativistic matter, that is, they push things apart and may even lead to

an accelerating expansion of the universe. Thus more components are present in

the cosmic fluid and thus our universe may be approximated by a perfect fluid

having many components, each with an equation of state of the form

Pi = wipi (2-1)with -1 3 w,- 3 +1, 2' = 1, 2, - - If we denote the total energy density due to all

such components as p, then

p:pm+Pr+Pv+' (2-2)where pm, p,., p,, etc. are the average densities of matter, radiation, vacuum

energy etc. In general,

p = 2 Pi: (2-3)1'

where p,-s represent energy densities of various components. From the energy­

momentum conservation law (here it is assumed that only the total energy density

is conserved), we have [14]

p = -33 (p+p) = —3§p(1 + w) (2.4)where p = Z,-pi is the total pressure and the ratio 111 = p/p should lie between

-1 and +1. Splitting p and p into individual components, the above equation

becomes

['71+/72+/'73+ = —3E[p1(l.-l-U11)+,02(1+’iU2)+"']Equating the RHS of equations (2.4) and (2.5), we will get

mm : * lP531(:/:3: "l _ 1 = Z1" Pi(:+ uh)

The conservation of individual components, which may be expressed as /3, =

-1 (2.6)

—3%p,~(1 + w,-), is only an extra assumption, since it does not follow from the

Einstein field equation. Equivalently, it can be stated that in a many component

fluid as in the above case, the Einstein equations, along with the equations of state

of individual components, are insufficient to determine the creation of individual

components, ie., individual ,1’),-’s. Thus it is more general to assume that the

total energy density is conserved, and this will lea.‘ to creation of one component

at the expense of other components (like particle creation from vacuum energy)

[55]. Since they are not uniquely determined by the field equations, such creations

can be considered sporadic events, like those occurring in active galactic nuclei,

which can result in fluctuations in the ratio p,-/ p. In [48], Weinberg discusses

some phenomenological proposals made by some authors, of the exchange of

energy between vacuum and matter, or vacuum and radiation, in such a way

that either p,,/pm or p,,/p, remains constant. He also considers the possibility

of creation of radiation from vacuum energy, keeping a fixed ratio p,,/pm. Here,

as in the case of other stochastic processes like Brownian motion, a complete

solution of the macroscopic system (the universe) would consist in solving all the

microscopic equations, describing the individual creation processes, but such a

rigorous derivation will be very complicated or even impossible. In this context,

a stochastic approach is more reasonable, in which we consider the creation rates

to be fluctuating, leading to fluctuations in the ratios p,-/ p. This, in turn, will

51

lead to a stochastic equation of state (fluctuating w), as can be seen from eq.

(2.6) above. Consequently, the expansion rate also will be fluctuating, and the

equation for the Hubble parameter will appear as a Langevin type equation (or

stochastic differential equation, SDE). Physically motivated interaction models,

which lead to energy transfer between various components, are proposed in the

literature [77, 81], but we propose this new phenomenological model to explain

the scatter in the Hubble diagram as arising from the possible fluctuations in

such energy transfer.

2.3 Stochastic approach to the standard model2.3.1 Stochastic Hubble parameter

Let us now formulate the stochastic theory, starting from the basic equation in

FRW model for the Hubble parameter: [13]

- 7G’H = —H'-’ — 4Tp(l + 3112) (2.7)

For the sake of simplicity, we assume that the effect of the curvature factor

appearing in the field equation is negligible, as in inflationary models, so that the

background is approximately flat. Hence above equation becomes

H = —§H2(1 + w(t)) (2.8)If 11) is a constant, we are back to the deterministic equation with solution H or

1/t. For the Friedmann dust approximation of the present universe, with w = 0,

we have H = 2/3t. With a fluctuating u". eq. (2.8) is a Langevin type equation,

describing the evolution of the stochastic variable H; ie., the fluctuating character

of w leads to a random behaviour (or evolution) for the Hubble parameter. We

52

use stochastic methods [82, 83] for the analysis of the above problem, and the

probability distribution function (PDF) of the stochastic variable is calculated

by solving the corresponding Fokker-Planck Equation (FPE).

To simplify the problem further, we make use of the transformation,

2

3H2: (2.9)so that eq. (2.8) becomes

z" = 1 + 111 (2.10)Here as is a measure of Hubble radius in the fiat FRW model. This equation is a

non-deterministic, stochastic, first order differential equation. When 11) = 0, this

equation is analogous to that of a particle moving in a medium with constant

velocity. With a fluctuating w, the analogous particle is subjected to random

forces as it moves. Now we make certain simplifying assumptions, which may

be stated explicitly as follows. We consider eq. (2.10) as a stochastic Langevin

equation with w as a Gaussian 6—correlated Langevin force term, whose mean

is zero. Though the assumption of 6—correlation and that of zero mean value

for w are taken for the sake of simplicity, we expect that they are reasonable,

considering the time-scales involved. The corresponding FPE, describing the

evolution of the probability distribution function W(I,t) is obtained from eq.

(2.10) by finding the drift and diffusion coefficients (from the Kramers-Moyal

expansion coefficients) [82].

2.3.2 The Fokker-Planck equation

If we have a general Langevin type equation [82] of the form

53

3) = h(y,t) + 9(1). t) W) (2-11)where l‘(t) is a fluctuating quantity, with zero mean and is Gaussian r5—correlated,

then the corresponding Fokker-Planck equation (FPE) describing the time-evolution

of the PDF, W(y, t) can be written as

aW+§§”‘—) = (—§y)nD<">(y>W<y,t> (2.12)where the D(”)(y) are the Kramers—Moyal expansion coefficients, which are gen­

erally defined by

D<"><y,t> =5 1irr5§<[y<t+T> -21") (2.13)- 7” y(i)==Here y(t + T) (T > O) is a solution of eq. (2.11) which at time t has the sharp

value y(t) = z. Under the assumption of 6—correlation and zero mean of F (t),

all coefficients vanish for n 2 3 and we retain only the coefficients D“) and D”),

called drift and diffusion coefficients, respectively. In the one variable case, the

drift and diffusion coefficients are

D"’(y, t) = 11(1). t) + %;’t)9(y, t) (2-14)and

D‘2’(y, t) = 92(1), 15) (2-15)In the present case these coefficients are found to be constants. The one variable

F PE [82] is

6W(y, t) 6 82?D‘2)(y) W(y. t) (216)

54

If we extend the procedure to the N—variable = y1,y2 - — - yN) case, then the

Langevin equation is

in = hi({y}3 1) + 9u'({y},t)1U'(t) (2-17)

with (F1-(t)) = 0 and (F,~(t)1"j(t')) = 6,-J-6(t — t’). The drift and diffusion terms

are

Di-({y},t) = hi-({y},t) + gu({y}, t)3im9ij({y},t) (2-18)

Du'({y}, t) = 9uc({y}a t)9jk({y}, t) (2-19)

and the FPE becomes

T = —1i(.,3y1_D£1’<{:/}>+ i 5§7jD£}’<{y}> W<{y},t). (2.20)‘=1 i,j=lThe FPE is just an equation of motion for the distribution function of flue­

tuating macroscopic variables. For a deterministic treatment, we neglect the

fluctuations of the macroscopic variables. For the FPE (2.20) this would mean

that we neglect the diffusion term. We come across such equations when dealing

with the Brownian motion of a particle. By solving the FPE one will get the

distribution function, from which averages of various macroscopic variables are

obtained by integration. Now let as return to the SDE (2.10). The corresponding

FPE can be written as

3W(W) : _D(1)3 +D<2>5_2at 8:1: 31:2We find the drift coeflicient D“) = 1, and the diffusion coefficient D9) = D, is

W(:r, t). (2.21)

assumed to be a constant. These coefficients follow from eq. (2.10). It will be

55

noted that this diffusion term arises from fluctuations in w alone. We can solve

the FPE by first assuming an ansatz

W(:z, .t) = ¢,,(a.-) e"\"’, (2.22)where we treat ¢,,(a:) and /\,, as the eigenfunctions and eigenvalues of the Fokker­

Planck operator

8 2

LFP = [——D(1)($) 6+ ,3::

with appropriate boundary conditions. Now we define two more functions in

order to get a solution for the FPE:

<I>($) — —/¥ 2;’ = -g, (2.24)and

Q —::/2D I —1,bn(a:) = exp ¢,,(z) = e qJ,,(:I:), (2.23)

where <I>(z:) is treated as a stochastic potential, and d1,,(a:) is an eigenfunction of

the Hermitian operator Ly

<13 <13L3 = exp(—)Lppexp(——). (2.26)2 2Making use of (2.22) and (2.25), the time independent part of FPE becomes

32¢n( ) 1 An61:22: = — ’l,(},,_(.'I7) = -‘k2’¢n(.’I,').

Here

An 1/2I: = i [3 — $] (2.28)

The most general solution to eq. (2.21) is

W(1:,t) = E c,,e"\"‘¢,,(:r), (2.29)n=0

where cn can be real or complex, but W(:c, t) is always real. When a stationary

solution exists, /\0 = 0. In the above situation, we see that for A" < 1/4D, k2

is negative and the solution 1,b,,(a:) is exponentially diverging, which is not an

admissible solution. Thus we conclude that An 2 5 so that Is is real, though

there is no stationary solution existing in this case. This is a constraint equation

for the eigenvalue parameter A,,, which has the dimensions of frequency. Thus

the only physically reasonable solution existing, is with

¢,,(::.-) = Aexp(ika:), (2.30)-00 < I: < +00, which gives

q5k(x) = Aexp +'ik1I,‘) (2.31)A is a normalisation constant. This situation is justifiable since it precisely cor­

responds to the deterministic solution 1 = t of eq. (2.10) when we calculate the

PDF One point we have to note here is that this eigenfunction formulation of

finding solution to FPE is applicable only if the drift and diffusion coefficients

are independent of time, as in our case. Following the standard procedure, we

make use of the completeness relation for 1,b(:c) to specify the initial condition

6<r — 2’) = /:° w;<x> was’) an: (2.32)We evaluate the transition probability for the stochastic variable to change from

the state :5’ at time t’ to I at time t as

57

P (a:,t | 3:',t') = exp (Lpp(t — t')6(:z: — 35)]

= exp " /:0 Wm) mm’) dk e'*“""1 — ’ — t — t’ 2= we, exp (2.33)2 7rD(t—t) 4 (t‘t)

The probability distribution functions at two times are related by

w(a:’,t’) = / P(x’,t’ |z,t) W(:r,t) da: (2.34)Since here the transition probability has the initial value

P(x,t | a:',t) = 6(a: — 1:’), (2.35)

the PDF W(1:, t) is the same as the transition probability. Thus we get the

distribution function as

_ 2\/7rDt ex 417??

which is Gaussian in form and is real. Note that we have chosen A = 1/ \/ 27r in eq.

Wm) — L p (_(i‘—’*)2] , (2.36)

(2.30) for normalisation purpose. VVe can immediately replace this distribution

function in terms of the stochastic Hubble parameter H as W’ (H, t). Dropping

the prime, we can write this PDF as

1 1 2 — 3Ht 2WW’ = mm “P (2-37)

The Gaussian in (2.36) has its peak moving along in such a way that the expec­

tation value of the variable is (:5) = t and this corresponds to the deterministic

solution of (2.10). The width of the Gaussian is found from the variance 02 =

58

((1 — (:1:))2) = 2 D t and 0 Z (1') till it = 2D. With H = 100 h kms‘1Mpc"1,

t =t17 x 10173 and D = D17 x 10173, the PDF can be written as

3.0856 1 (6.1712 — 3m”)?W h_ = T ____j( I 3,12 \/7TD]_7t17 exp 36h.2D17t17

For the range of values of interest, 1 < tn < 5 and D17 ~ 10‘3, W(h,t) is

(2.38)

approximately a Gaussian.

2.4 Comparison with data and conclusions

In section 2.2, we have shown that fluctuations in the creation rates are physical

processes which can lead to a stochastic equation of state. A fluctuating w factor,

in turn, will lead to fluctuations in the time evolution of the Hubble parameter;

ie., the expansion rate of the universe becomes a stochastic quantity, instead of

remaining a deterministic variable. We argue that such a fluctuating expansion

rate might have led to randomness in the recession velocities of objects (galaxies

and other extra galactic objects) in addition to peculiar velocities. The effect

of this randomness of recession velocities is similar to that of peculiar velocities,

and will produce scatter in the Hubble diagram.

From the Hubble diagram, one can find the PDF for the present Hubble

parameter I-I0, which arises from the point-to-point variance of the measured

Hubble flow, in the following way. We use the data of 42 Type Ia supernovae

(the supernova cosmology project) given in a paper by Perlmutter et al [25].

The traditional measure of distance to a supernova (SN) is its observed distance

modulus pa = mm — M501, the difference between its bolometric apparent and ab­

solute magnitude. In FRW cosmology, the distance modulus is predicted from the

source’s redshift 2, according to up = 5 log (fit) +25, where DL = r,~a(t0)(1+z)

is given by equations (1.103) and (1.104) for the present model. Thus one can

59

obtain the predicted distance modulus of an object with redshift z. Convention­

ally, assuming that the observed and predicted distance moduli coincide, one can

find a value of Ho. For a collection of objects, one can find the likelihood for Ho,

from a X2 statistic:

(ll ,1‘ — #a,z')2X2 = Z —p——2— (2.39)1' 0:‘where 0, is the total uncertainty in the corrected peak magnitude of SN Ia, which

includes dispersion in galactic redshift due to peculiar velocities, uncertainty in

galaxy redshift and other effects. For the special model we are considering, h is

the only parameter and the normalised PDF can now be obtained as (we use the

methods given in the paper by Riess et al. [26])

pm I u.) = (2.40)The Hubble constant, as derived by these authors [26] from the MLCS and TF

approaches, are 65.2d:1.3 km s‘1Mpc‘1 and 63.8i1.3 km s‘1Mpc“ respectively.

This, they claim, are extremely robust and do not include anysystematic errors

like the uncertainty in the absolute magnitude of SN Ia. In the present case, we

have computed p(h | pa) (where Ho = 100h km s‘1Mpc‘1) using the supernova

data in [25], which corresponds to their Fit C and attempt to compare p(h | pa)

with the theoretical PDF W(h, to) of the present Hubble parameter, to evaluate

the diffusion constant D appearing in the expression. It is found that the two

curves, shown in Fig. 2, coincide for a value ofD = 3.77 x 10135 (This corresponds

to an age 3.029 x 10175). The best fit value is h = 0.679 and a 68.3% credible

region has a half width 0;, = 0.011. Since our primary objective is to make an

order of magnitude evaluation of D, we choose a fiducial absolute magnitude for

60

SN la in computing no, equal to -19.3 mag. Slight variation in this quantity will

not significantly affect D, though the best fit value for h may change.

The stochastic nature of the expansion rate provides a cause of systematic

error in measuring the true value of the Hubble constant Ho, apart from the

peculiar velocities induced by the observed density fluctuations. The conventional

explanation may be adequate to account for the observed peculiar velocities of

objects, in the range of 100 km s“1Mpc‘1 — 400 km s‘1Mpc'1. The scatter

in the Hubble diagram at low redshifts may be explained on this basis. But,

since the amplitude of density fluctuations in the early universe was very low,

the scatter at very high redshifts remains unexplained, and it is desirable to look

for some alternative mechanisms which can induce random motions, like the one

presented here.

Moreover, since a fluctuating w factor (in the equation of state) can lead

to fluctuations in ,0, our stochastic approach can be extended to include the

density fluctuations also. We will discuss in detail the stochastic approach to

the evolution of cosmological parameters including p in the next chapter. Once

it is possible to identify some standard candles for quasars, we can apply this

formalism to estimate the value of the diffusion constant D. A stochastic theory of

density fluctuations also helps us to find this value from observations (if provided

with the data), and to check whether these different estimations give identical

results. If there are some explicit examples of models where a stochastic w

emerges, the predicted value of D may be compared with our estimation, but

here, we have not made any attempts in this regard. Thus the scatter in the

Hubble diagram is an indication of non-deterministic behaviour of H, whose

randomness is explained on the basis of a stochastic theory.

61

40 I I I I I I I7 I I I’p(h)' »an ’H(h,t.)'

.4 «R35 - «r ". ­i’ ‘Af A]q i3°- l L. ‘._' i25 - f ‘ ­1 1E 4 ‘l Ja: 2°’ 4 t

J.73 ‘J '3‘U3’ 15 ' .« J. 'E «g 5’:lg 10 - , 3 ­a d 3~' 15 - :7’ 1 .J" *29 Q1 L I | n I A n 1 10.68 0.53 0.7 0.72 0,74 0.78 0.78 0.8

h

o

0.58 0.5 0,152 0.54

Fig. 2 The PDF vs h. The dotted line is the PDF calculated using observational data andthe solid curve is the theoretical one, obtained by plotting W(h,t)

Chapter 3

Evolution of cosmologicalparameters in the new model

3.1 Introduction

In this chapter we develop a stochastic formulation of cosmology in the early uni­

verse, after considering the scatter in the redshift-apparent magnitude (z — m)

diagram in the early epochs as a piece of observational evidence for a non­

deterministic evolution of the early universe. The standard model is based on

the assumption of homogeneity and isotropy of matter distribution on very large

scales (cosmological principle), which leads to the interpretation of deterministic

and linear Hubble’s law, and one expects a scatter-free z — m diagram for galax­

ies and other extra galactic objects. For nearer galaxies (low 2), the scatter is

small, and this can be accounted by the conventional peculiar velocities. How­

ever, the scatter increases with z and the large randomness observed for high

redshift objects (early epochs) is not due to peculiar velocities alone, but due to

some other mechanisms. In the preceding chapter we proposed that a stochas­

tic equation of state or a fluctuating mean 112 factor in the equation of state led

to a non-deterministic or stochastic Hubble parameter and argued that such a

fluctuating expansion rate in the early universe might have led to randomness

62

in the recession velocities of objects, in addition to peculiar velocities, and will

produce the scatter in the redshift-apparent magnitude diagram in those epochs.

The other consequences of such a fluctuating expansion rate will be discussed

in the concluding chapter. Here we formulate a more general description of the

stochastic dynamics of the early universe in the Fokker-Planck formalism and

discuss the non-deterministic evolution of the total density of the universe. Since

the evolution of the scale factor for the universe depends on the energy density,

from the coupled Friedmann equations (modified) we calculate the two-variable

probability distribution function (PDF) using the Fokker-Planck equation (FPE).

Such a stochastic approach is necessary when the mean w factor in the equa­

tion of state of the cosmic fluid is a fluctuating quantity. We can say that the

evolution of the universe in those epochs is non-deterministic (becomes nearly de­

terministic for the present epoch) and the corresponding dynamical equations are

of Langevin type where one can evaluate the PDFS of the stochastic variables. We

also make clear that the fluctuations in the ratio p,- / p (which leads to a stochastic

equation of state) that we are taking into account here (see section 2.2), are clas­

sical; ie., our stochastic model is a modification to the classical Friedmann model

of the early universe (2 ~ 1 to 10 and may be even higher), when fluctuations are

significant. In [84, 85] Fang et al. discuss a stochastic approach to early universe

(before the recombination epoch), where the cosmic fluid consisting of primeval

plasma and radiation, is not perfect, but has dissipations due to differences in

the adiabatic cooling rates of the components of the cosmic fluid and the possible

energy transfer between them. However, once we probe into still earlier epochs

(stages of inflation etc.), quantum fluctuations become very important. Many

authors [86-89] discuss the need for a stochastic approach to inflation, when the

quantum fluctuations of the scalar field are significant, and try to get a PDF for

63

the scalar field after solving the quantum Langevin equation (or FPE) describing

the evolution of the scalar field. However we adopt the stochastic approach in

the classical regime, where fluctuations in the creation rates and also in the pos­

sible energy transfer between different components of the cosmic fluid, lead to a

stochastic equation of state. This causes a non-deterministic (stochastic) expan­

sion rate for the universe, and the dynamics of the early universe is described by

a set of stochastic differential equations instead of the deterministic Friedmann

equations of standard model.

3.2 Stochastic evolution of the cosmological pa­rameters

3.2.1 Density parameter

Suppose the universe is approximated by a many component fluid in the early

epochs, with a fluctuating w term in the equation of state. Now we write the

evolution equation for the total density in those epochs (assuming that total

energy density is conserved), immediately after inflation, when the curvature

factor appearing in the field equation is negligible. Therefore,

[1p = -3 [1 + w(t)] p. (3.1)0.

Using Friedmann equations [14] we have

/2 = —24,—.G [1 + w(t)] p3/2 (3.2)

This equation is a SDE of the Langevin type. Since in is a fluctuating ‘force’

term, p is a stochastic variable, ie., its evolution is non-deterministic. The random

behaviour of p in the early universe is due to fluctuations in the factor 11; alone. If

64

fluctuations are zero, we are back to the deterministic standard model. We apply

the standard stochastic methods to this equation and the PDF is calculated using

the FPE.

By making use of the transformation,

1= , 3.30 ,T P ( )eq. (3.2) becomes

a = 1+ w(t), (3.4)which is again a non-deterministic, Langevin type equation, we come across such

equations in Brownian motion etc. (here 0 oc t for a pure deterministic situa­

tion as in the standard model). To solve eq. (3.4) we use certain simplifying

assumptions we already made in section 2.3.1. The F PE formed from eq. (3.4) is

6W(o, t) _ [ 6 62at —‘a—c)_’+D¥:| l/V(0',t).Here again diffusion coefficient D”) = D (D is a constant with dimensions of

time, which is introduced for the purpose of generality). In order to obtain

non-stationary solutions of eq. (3.5), we use the following separation ansatz for

W(a, t),

W(0, t) = ¢(a) exp(—/\t). (3.6)Substituting this into (3.5) and solving for ¢(a) we get

45(0) = Aexp [% + ikcrj , (3.7)where

65

/\ 1Rim3 _D 4132. (3.3)Thus we see that for A < 1 / 4D, 1:2 is negative and the solution is exponentially

diverging, which is not physically admissible. Hence we conclude that A 2 1/4D,

so that k is real. We write the most general solution as

W(a, t) = Z c:,,qbn(0) exp(—/\nt), (3.9)

where c,,_ can be real or complex. For a continuous parameter k, from equations

(3.6) and (3.7) the general solution or the distribution function is given by

0'

W(a,t) =A[_:°exp 2D +130 — 3201- 1% dk. (3.10)

We choose A = 1/27r as a normalisation constant. On evaluating the above

integral, we get the PDF as

W(cr,t)= 1 exp[:fl, (3.11)x,/47rDt 4D15

which is Gaussian. The average value of the stochastic variable cr is (0) = t

(corresponds to the deterministic situation). The variance 1) = ((0 — (a))2) =

2Dt. Once W(cr, t) is known, it is straight forward to write the distribution

function W(p, t) as

W(p, t) = ‘(I ' t” 6”C”)2] (3.12)1

\/967r2DGp3t exp [ 247rGDpt

We can also find the transition probability using an equation similar to (2.33) for

the stochastic variable to change from an initial state (a', t’) to a final state (a, t)

BS

66

P(a, t 1 a’, t’) = 4flD1(t_ t’) exp , (3.13)with the initial value

P(a,t | a',t) = 6 (0 — 0"), (3.14)indicating a Markovian nature [82] for the stochastic variable 0 and the PDF

W(z,t) is same as the transition probability P(a:,t | 0,0). In terms of p eq.

(3.13) becomes

. , _ 1 ‘l(\[_I'\/5)‘\/6770!?/9’(t‘t,)l2P(p’tlp’t) — 47rD(t—t')exp 247rGpp'D(t-t’)

(3.15)

This represents the probability for the energy density to change from an initial

value p to a final value p’ during a time interval (t — t’) in the early epochs. This

characterizes the stochastic behaviour of density evolution in the early universe.

3.2.2 Scale factor and density parameter together as atwo variable Fokker-Planck problem

Under the assumption that the factor 112 is fluctuating during the early epochs,

the evolution of the scale factor also becomes non—deterministic, since the time

evolution of a(t) is determined by the total density of the universe. So we have

a system of coupled SDEs derived from Friedmann equations [14]. We have

(1_ 87rGpE——‘/ 3 , (3.16)

67

and

p = —\/2'-'17rG'[1 + w(t)]p3/2 (3.17)

Here we are considering the dynamics of the universe after the inflationary stage.

With the transformation (3.3), the above system of equations reduces to

(1 = §—:, (3.18)and

(7=1+'w(t). (3.19)Following the standard procedure [82] (consider the equations (2.17) to (2.20))

we have the drift coefficients D9) = 20/30, D31) = 1 and the diffusion coefficient

D352 = D is assumed to be a constant. It may be noted that the diffusion term

arises due to fluctuations in w alone. The two variable FPE for the distribution

function W(a, 0', t) can be written as

BVV 2 3W 3W 32W_ = __ j _ Z Z _23t 30 lw a 3a l 60 D 302 (3 0)To solve this equation, we assume an ansatz

W(a,0,t) = U(a) V(a) exp(—/\t) (3.21)

and substituting into eq. (3.20), we obtain

0' d2V 0 dV 2 a dU— —— — — — = — — — 3.22VDda2 Vda+)‘0 3lUda+] ( )Each side of this equation can be equated to a constant m, since the LHS depends

only on 0 and the RHS only on a. When m = 0, ‘—iUQ oc —d—'’, which on integration(1

gives

68

U(a) oc 1, (3.23)and

V(0) oz exp 2D

with it given by eq. (3.8). A physically reasonable solution exists for A 2 1/4D,

1 + ika] , (3.24)which is

BW(a, 0, t) = 3 exp 2% + ika — At] (3.25)

Here B is a normalisation constant, chosen to be 1/27r. One point to be noted

is that, the most general solution to eq. (3.22), when m ¢ 0, is a series solution

owing to the essential singularity at 0 = 0. One can find a limiting solution as

0 —) O, in the following form

2m. X (L) °° (ma/D)"W“‘*”)*““‘However, we will get a real general solution in a compact form after integrating

(3.26)n.=1

eq. (3.25) in the range -00 < k < +00,

W(a, 0', t) = \/£3? (a)—1 exp [— (U4_D:) J (3.27)In terms of p, it becomes

W(a, p, t) = (3.28)\/967r7GDp3a.2t 247rGDpt

The two variable PDF is Gaussian in 0, and diverges as a —> 0 , where classical1 exp [_(1—t\/67rl§p)2

approach fails and quantum theory [90] takes over. Now we write the expression

for the transition probability (see equations (2.33) and (2.34)):

69

In terms of p and a it becomes

, , , _ (a')“1 _l(\/_,_\/5)-\z67TGppI(t_t,)l2P(a’p’t|a’p’t) - \/47r;(t—t’)expl 247rGpp'D(t-t’)

(3.30)

which represents the transition probability for the variables to change from the

state (a', p’) to (a, p). Thus the scale factor a together with the density p evolves in

a non-deterministic way, which.in turn, strongly influences the formation of large

scale structures in the universe, since the evolution of the density perturbations

also depends on w. This we consider in detail in the final chapter. In all these

cases we get Gaussian distributions, which are sharply peaked initially, but spread

out with time.

3.3 Conclusion

To conclude, we note that the stochastic approach presented in the last two chap­

ters is a modification to the standard model, when fluctuations are present. In

the standard model where the cosmological principle is strictly valid, we have a

deterministic evolution of the universe. However, when we probe into the early

epochs (possibly after inflation), the dynamics of the universe can not be de­

scribed by the purely deterministic equations of Friedmann, since fluctuations

are present. The observations show a scatter diagram for redshift-magnitude

relation for supernovae and high redshift objects. This scatter diagram is an

indication of the non-deterministic behaviour of the Hubble parameter, whose

randomness is explained on the basis of a stochastic theory, after introducing

70

Chapter 4

Application of the stochasticapproach to the generalizedChen—Wu type cosmologicalmodel

4.1 Introduction

In this chapter we reconsider the classical stochastic model of cosmology devel­

oped in the last two chapters. The uncertainty in the determination of the Hubble

parameter, which is a measure of the expansion rate of the universe, is one of the

most intriguing issues in the history of cosmology. The origin of uncertainty is

obvious from the redshift-apparent magnitude diagram; despite rigorous attempts

to control random errors in measurement, there is a clear scatter in it, though it

is now possible to narrow down this to a great extent. But, now we will show that

by using the z — m data for Type Ia supernovae [25], the scatter increases as we

go to higher redshifts. In Chapter 2, we have attempted to explain this scatter

as arising from an inherent stochastic or non-deterministic nature of the Hubble

parameter. It was shown that a fluctuating u'—factor in the equation of state

p = wp will lead to this kind of behaviour for H. and the equation for the Hubble

parameter will appear as a Langevin type equation. There we assumed for the

T3

sake of simplicity that space sections are flat and w is a Gaussian 6—correlated

stochastic force with zero mean. With these assumptions, we have written the

FPE, whose solution gives the theoretical PDF for H0 at time to, denoted as

W(h, to) (where Ho = 100h km s‘1Mpc‘1; The subscript 0 denotes the present

epoch). Using the z — m data no (0 corresponds to observational distance modu­

lus) for SN Ia used in [25, 26], we computed the observational PDF p(h | no) for

h in the present universe, again assuming its space sections to be flat. This PDF

arises from the point to point variance of the Hubble flow. We compared the two

plots for the present universe (see Fig. 2) and found them to agree well, for a

value of the diffusion constant, appearing in the FPE for the stochastic Hubble

parameter, equal to 3.77 x 10133.

This result is a first step towards an understanding of the anomalous scatter in

the Hubble diagram at high redshifts. However, there are certain refinements to

be made in our analysis. One drawback of the above scheme of comparing these

two PDFS is that when we derived W(h, t), the assumption was made that to has

mean value zero, whereas the observational PDF p(h | pa) was evaluated for a

model which contains matter and vacuum energy, which has mean total pressure

negative. Instead, if we had used in this evaluation the expression for the distance

modulus for a flat universe, which is matter dominated (ie., with w = 0), an

observational PDF would have been obtained, but the best fit value for It would

be ridiculously low. But most of the present observations are incompatible with

an QA = O flat model.

Another shortcoming is that though in both cases we take the PDF for h,

it remains to be explained how legitimate is the comparison of W(h, to) for the

present universe with a PDF p(h | pa) evaluated using the data that include high

2 objects, which belong to the distant past.

74

Now we try to rectify these two defects and to make a more rigorous test of

the stochastic assumptions using observational data by (1) comparing both the

theoretical and observational PDFs evaluated for the same model, which is an

alternative flat model [91], and (2) evaluating the observational PDF p(h,- | #0,)

for the Hubble parameter at the same epoch tj as that in the theoretical PDF

W(h,-, tj). This procedure helps us to compare the theoretical and observational

PDFS for the Hubble parameter for the same model and at the same epoch.

The value of the diffusion constant evaluated at any time is obtained as nearly

a constant, in agreement with our assumptions. A novel feature in our new

approach is that we evaluate the observational PDF for the Hubble parameter

at various instants in the past, also with an objective of justifying our assertion

that the scatter increases as we go into the past.

4.2 Stochastic approach to the new model

In all FRW models, the Einstein equations, when combined with the conservation

of total energy density, can be written in terms of the Hubble parameter as

H = —H2 — £7%g(p+ 312). (4.1)

If we restrict ourselves to flat models, then (with p = wp),

H = —§H2(1 + w). (4.2)In Chapter 2, we considered this flat case and assumed that w is a Gaussian

6—correlated Langevin ‘force’ term with mean value zero. This means that the

mean total pressure of the universe is zero, the same as that for dust. But

many recent observations [68-80] are incompatible with this model and hence, as

75

mentioned in the introduction, we look for a more observationally correct, but

simple model to apply our stochastic approach.

4.2.1 Generalized Chen-Wu type cosmological model

The deterministic model [91] we propose to use is the one in which the total

energy density obeys the condition p + 3p = 0, and hence having a coasting

evolution (a oc t). On the basis of some dimensional considerations in line with

quantum cosmology, Chen and Wu [50] have argued that an additional component

which corresponds to an effective cosmological constant A, must vary as 1/a2 in

the classical era. Their decaying-A model assumes inflation and yields a value

for qo (deceleration parameter). Their model alleviates some of the problems

of the standard model, but their results were found to be incompatible with

observations. In [91], the authors generalize this model by arguing that the Chen

-Wu ansatz is applicable to the total energy density of the universe and not to

A alone. If we assume that the energy components in this model are ordinary

matter and vacuum, then the condition p + 3p = 0, gives pm/p,, = 2 and if it

is only radiation and vacuum, then p,/p,, = 1. In [91], it was shown that in

this model, most outstanding cosmological problems such as flatness, horizon,

monopole, entropy, size and age of the universe, cosmological constant etc. are

absent. It was also shown that this model can solve the problem of generation of

density perturbations at scales well above the present Hubble radius and that it

can generate such density perturbations even after the era of nucleosynthesis.

4.2.2 Evolution of the Hubble parameter in the newmodel

The new deterministic model has 111 = -1 / 3 in the deterministic case, we rewrite

eq. (4.2) with w’ = w + %, as

76

- 3 2 ,H=—— 2- . .2H(3+w) (43)Now we assume that w’ fluctuates about its zero mean value and is 6—corre1ated.

Making the substitution

(4.4)

the above equation becomes

_ 3When 11)’ = 0, this is a deterministic equation, and the solution is straight forward.

However, with a fluctuating w’, the variable :1: becomes stochastic and one can

find only the PDF of such a variable. This can be done through the Fokker-Planck

formalism as in the previous cases. With the drift coefficient D“) set equal to

unity (follows from the above equation), and the diffusion coefficient D”) = D

being assumed to have some constant value, to be determined from observations,

we can write the FPE as

8W'(:r, t) 8 82 ,—— = —— — W . .at [ 0:: + 69:2] ($’t) (4 6)To solve the FPE, we assume the ansatz:

W'(x,t) = <;b,,($) exp(—A,,t). (4.7)

The remaining procedure to find the PDF is the same as that described in section

2.3.2. The transition probability for the variable to make a transition from (:r' , t’)

to (a:, t) is

77

P(z,t | z',t') = ¢—:exp [— (4.8)For the special initial value

P(:c,t 1 :13’, 4) = 5(:r — z’), (4.9)

the transition probability P(z,t | 1:’, t’) is the distribution function W'(:c, t). In

our case, we have the initial condition 2: = 3' = 0, at t = t’ = 0, so that

(4.10)W’(a;,t) = P(z,_t 1 0,0) = (“T ' 02]1

2)/7rDt exp i_ 417$

This also a Gaussian distribution function with (I) = t. In terms of stochastic

Hubble parameter (in the new model), the distribution function becomes

fl,/FD; _ 4 H20:With H = 100h km s‘1Mpc‘1, t = t1-, x 10173 and D = D17 x 10173 the PDF

W/'(H,t)= 1 $exp[ (4.11)

W(h, t) can be written as

W(h, t)3.0856 1 3.0856—ht 2

= exp [— (——-1] (4.12)2 I12 V 7l'.D17t17 4 h2D17 tn

For the range of values of interest, 1 < tn < 5 and D17 ~ 10'3, W(h, t) is

approximately Gaussian. For fixed D, the half width of the Gaussian is found to

increase as we go to lower values of t.

4.2.3 PDF for H from observational data

Conventionally, assuming that the observed and predicted value of distance mod­

uli coincide, one can make an estimate of the present Hubble constant, where

78

filo = mbal - Mbol:

and

D

up = 510g [IMZC] + 25.

The luminosity distance is D; = rj a(t0) (1 + z) (a(t0) is the present scale factor

and Ti is the radial coordinate of the SN Ia which emitted the light at some time

tj in the past). Here

[to dt”‘ tam‘For the coasting model ((1 oc t) discussed in the previous sections, for curvature

k = O, rj can be evaluated as

_ to t0 dt _ to7'3‘ — E ti Y — $;)'lI1(1+Z),

so that

_ (1+z)Ho

1M1+z) (in)One can substitute this into the expression for up to obtain the predicted distance

modulus of an object with redshift z. For a collection of objects, the likelihood

for H0 can also be found from a X2 statistic (see eq. (2.39)). The normalised

PDF for h can be found from [26]

exp(-X’/2)ff: dh exp (—X2/2)

As in Chapter 2, we compute p(h | ,u,,) for the new model, using the SN Ia

WI | Ho) = (4-15)data in [25], which corresponds to their Fit C and attempt to compare p(h | ya)

79

with the PDF W(h,to), to evaluate the diffusion constant D appearing in this

expression. It is found that the two curves, shown in Fig. 3, coincide for a value

of D 2: 2.36 x 10133 (This corresponds to an age 4.8583 X 10173). Here also we

choose the absolute magnitude appearing in the expression for observed distance

modulus to be -19.3, a slight variation in this quantity does not significantly

affect D.

In the above, we compared the theoretical and observational PDFS for the

same alternative model and thus it does not have the first shortcoming mentioned

in the introduction. The other incompatibility which still exists can be explicitly

stated as follows: W(h, to) is the PDF for the Hubble parameter of the present

universe, and it contains the diffusion constant D. But p(h 1 pa), which we try

to identify with W(h, to), depends on the scatter in the Hubble diagram for all

ranges of 2. For instance, if we include more high redshift objects in our sample,

the scatter would be larger and hence the half-width of the distribution p(h | pa)

will be larger. This, in turn, will affect the computed value of D, which is quite

unreasonable.

This problem can, however, be overcome if we agree to compute p(h_.,- | #0,) for

each value of redshift z (or for small enough redshift intervals centred about such

values), and compare these with W(h,-,tj) that corresponds to the same epoch

tj. To do this, we modify eq. (4.14) by re-evaluating rj in (4.13) in a different

way. One can also write, for the new deterministic model

. r d 1Tj = t] =so that

80

DL = (1;?l21n(1+ 2) (4.17)Evaluating up using this expression, we can evaluate X2 and hence also p(h,- | ,u,,,-),

which is the PDF for the Hubble parameter at some particular value of 2. We di­

vide the data in [25] for various redshift intervals around 2 = 0.05, 0.15, 0.35, 0.45,

0.55 and 0.65, each with A2 = 0.05. The PDF for the average Hubble parameter

for such intervals is calculated with an expression identical to (4.15). The results

are plotted in Fig. 4 along with the corresponding theoretical PDF W(h,-,t,-)

which overlaps with them. The relevant parameters are given in Tablel.

4.3 Conclusions

It is noted from Fig. 4 and Tablel that, for the intervals with larger values of z,

the 68.3% credible region of p(h | no,-) has a halfwidth ah, which also increases.

This behaviour is the one expected from theory, as noted while plotting the

theoretical PDF (4.12). Physically, this means that the scatter increases as we go

to higher redshifts. The intervals with centre at 2 = 0.15 and 0.35 are exceptions

to this, but this may be due to the fact that these intervals contain only very

few objects. As more SN Ia are observed in these redshift intervals, an accurate

picture will emerge.

The value of the diffusion constant D evaluated for various intervals, however,

does not show any dependence on 2:. This justifies our assumption that D is some

constant.

A pitfall, even in the present analysis, is that the intervals we consider are

with A2 = 0.05 and this value may not be small enough to give correct answers.

This, again, can be overcome only in the future, when the number of observed

SN Ia becomes large.

81

This chapter is a modification to the theoretical investigation on the origin

of the random motions that cause large scatter in the Hubble diagram at high

redshifts. Conventionally, the random motions are viewed as peculiar velocities

induced by the observed density fluctuations. Given the fact that density fluctu­

ations are evolving phenomena, peculiar velocities induced by them can not lead

to the large randomness observed at early epochs, though it is a feasible phys­

ical process in the late universe. A fluctuating expansion rate arising from the

stochastic nature of the equation of state, on the other hand, provides a natural

explanation for the large scatter in the Hubble diagram at high redshifts.

82

Table 1. Diffusion constant for various epochs

Redshift No. of SNe Best fit _Standard Age in Diffusionz in the value of deviation units of 10” s constant

interval 2 :1: 0.05 h 0;, tn D s0.05 15 0.693 0.011 4.4502 0.5775X 10140.15 3 0.772 0.025 3.9987 2.147 X10140.35 5 0.830 0.024 3.7204 1.5 X10”0.45 15 0.875 0.017 3.528 1.655 X10”0.55 7 0.985 0.026 3.1345 1.12 X10”0.65 6 1.043 0.030 2.959 1.226 X10”

83

70 If I I I I I I I I T'na.a_|-lhot’ o

A 'neu_ph’

-v-w-aha

probabl 111:9 distribution Function 0 . . . L . J . .10,48 0.5 0.52 0.54 0.58 h 0.58 0,8 0.62 0.84 0,65 0,88 0.7

Fig. 3 Observational and theoretical PDFs vs h, using the redshift-apparent magnitude (2­m) data for Type Ia Supernovae as given in [25], which corresponds to their Fit C. Thecontinuous line is for the theoretical PDF, whereas the dotted line gives the observationalPDF.

4'0 I I I I I IA35 - ­Z=Cl,05

i ’:‘ 925- I f ­5 is 3 9‘§ ’ z=o.45

E 20 ­§ ..*3 *3 15 - f’+2 23 :3 1°­3§51 5' ifV:o ' ‘ ‘ " _» ..0,5 0,8 0.7 0.8 [U 0.9 1 1,1 1.2

Fig. 4 Observational and theoretical PDFs vs h for various epochs centred about2 = 0.05, 0.15, 0.35, 0.45, 0.55, 0.65, using the z-m data for Type Ia SNe as given in [25],which corresponds to their Fit C and which lies in the interval A2 = z i 0.05

Chapter 5

Other problems with thestandard model and discussions

5.1 Introduction

The basic assumption of the standard cosmological model is Einstein’s cosmo­

logical principle which, in fact, is the hypothesis that the universe is spatially

homogeneous and isotropic on large scales. It is thus assumed that the large

scale 3D‘ geometry of the universe is isotropic and homogeneous. Many of the

problems of the standard model are a direct consequence of such simplifying as­

sumptions. The cosmological principle implies linear deterministic Hubble’s law,

12 = Hr (or a linear, scatter-free redshift-apparent magnitude relation), which

is valid at scales where matter distribution can be considered on an average

uniform, and is well established within local scales (Since the early 1980s, multi­

object spectrographs, CCD detectors etc. have allowed the mass production of

galactic redshifts). However, many recent analyses (redshift surveys such as CfA,

SSRS, LEDA, IRAS, Perseus-Pisces and ESP for galaxies and Abell and Aco for

galactic clusters) have revealed that the three dimensional distribution of galaxies

and clusters of galaxies are characterized by large-scale structures (hierarchical)

and huge voids [33-42]. Such a distribution shows fractal correlation upto to

84

the limit of available samples. This has confirmed the de-Vaucouleurs power-law

density-distance relation [33],

pm oc (f—0)D1—3 (5.1)with fractal dimension D1 2 2 at least in the range of scales 1 to 200 h‘1Mpc,

ie., a sheet-like distribution of galaxies. In the above expression D1 = 3 cor­

responds to perfect homogeneous distribution of galaxies and a perfect linear

Hubble expansion.

A fractal [29] is a geometric shape that is not homogeneous, yet preserves the

property that each part is a reduced scale-version of the whole. That is, fractals

are self-similar structures or possess scale-invariant properties. If the matter in

the universe were actually distributed like a pure fractal on all scales, then the

cosmological principle would be invalid, and the standard model in trouble.

Thus the universe is inhomogeneous and essentially fractal-on the scales of

galaxies and clusters of galaxies, but most cosmologists believe that on much

larger scales it becomes isotropic and homogeneous [37, 38], eventhough the cross­

over scale to homogeneity is not yet identified [33]. According to the standard

model of cosmological structure formation, such a transition should occur on

scales of a few hundred Mpc. The main source of controversy is that the most

available three—dimensional maps of galaxy positions are not large enough to

encompass the expected transition to homogeneity. Distances must be inferred

from redshifts, and it is difficult to construct these maps from redshift surveys,

which require spectroscopic studies of large numbers of galaxies. Sylos Labini

etal. [42] have analysed a number of redshift surveys and find D1 = 2 for all

the data they look at, and argue that there is no transition to homogeneity for

scales upto 4000 M pc, way beyond the expected turn over. A controversy exists

85

among cosmologists regarding this switch over scale to homogeneity. The fractal

behaviour of galaxy distribution within a scale of ~ 200 h"1Mpc (this scale may

be even deeper), is a challenge for standard cosmology, where a linear Hubble’s

law is a strict consequence of the homogeneity of the expanding universe. The

presence of dark matter (distributed homogeneously) may save the cosmological

principle even at small scales [39]. In this way, one may save the usual FRW

metric (which needs a homogeneous density), while a substantial revision to the

models of galaxy formation is required. On the contrary, if the dark matter is

found to have the same distribution of luminous one, then a basic revision of

the theory must be considered. In fact, from a theoretical point of view, one

would like to identify the dynamical processes which can lead to such a fractal

distribution

5.2 A stochastic evolution of density perturba­tions

The structures we see today are formed by a process known as gravitational in­

stability, from primordial fluctuations in the cosmic fluid [15, 16]. But, because

the strength of clustering is expected to increase with time (evolution of density

contrast being proportional to some power of scale factor in the linear approx­

imation according to the standard model), the galaxies must deviate from the

smooth Hubble expansion. These deviations away from uniform Hubble’s flow

are known as peculiar velocities. According to the standard Friedmann model,

6v oc Q8'56p, where 6p is the density perturbation and (20 is the present value

of the ratio between critical density and density of the universe. This peculiar

velocity is one of the independent probes of inhomogeneities in the gravitational

field, induced by the density fluctuations. Another probe is the fluctuations in

86

the background radiation. Observations show a very nearly linear Hubble expan­

sion for local scales, and deviations from this deterministic Hubble’s flow increase

with redshift, as is obvious from the Hubble diagram, which is a scatter diagram

and the scatter increases as we go to early epochs [20, 24, 28]. The conventional

explanation for the scatter is in terms of the peculiar velocities alone, induced by

observed density fluctuations. But density fluctuations are evolving phenomena,

they cannot induce the large randomness observed at high redshifts. The scatter

in the Hubble diagram or deviations from the linear Hubble expansion may also

arise from an inherent stochastic nature of the Hubble parameter, apart from the

peculiar velocities which are significant only in the late universe. We have ex­

plained the anomalous scatter in the Hubble diagram at high redshifts on the basis

of a fluctuating or random expansion rate of the universe, thanks to a stochas­

tic equation of state. Under a stochastic equation of state, dynamical equations

describing the evolution of density and its perturbations must be stochastic and

only a PDF can be found for these quantities. The stochastic evolution of density

and other cosmological parameters are described in the previous chapters. Here

we will show that under the above circumstances, the time-evolution of density

perturbations is described by Langevin type equations

In the early universe, when cosmic fluid is not uni-component, a stochastic

equation of state emerges. ie., the in factor becomes a fluctuating quantity. Hence

the evolution of density perturbations is a stochastic process. In the following

we will show that the PDF of the stochastic density contrast is approximately

Gaussian. In the early universe, the energy density in any region can be written

as a perturbation equation [16]:

plus, t) = mt) + ape, t), (5.2)87

where pb, is the background density, which at any time t is independent of lo­

cation. However, p at different regions is slightly different in the early universe,

and hence 6p also. The evolution of density contrast (6 = 6p/pb) according to

FRW model, is a deterministic one, proportional to some power of the scale fac­

tor in the linear approximation. In the stochastic approach, due to a fluctuating

equation of state, its evolution is a stochastic process. We assume that the total

energy density is conserved, so

:3 = -3H(p + p)- (5-3)After some simplifications, we get the evolution equation of 6p [16, 17] in the

form

6/) = —3(Pb+Pb)5H — 3H,,6p. (5.4)

Here the suffix b stands for the background. Using 6p = p56, the above equation

becomes

6 = 3Hbw(t)6 - 3[1+w]6H. (5.5)

Now using the relation

H? = $21. (5-5)we have

(H — 6H>"’ = 8—’;9<p — cm. (5.7)

Equating both sides, we have

88

and

2H6H = gigqép.

Using equations (5.8), (5.9) and (5.2), we get

1 87rG 6p -1/26H = —(/——6 '1/2 [1 + —]2 3 P (Pb) pb

Retaining only first order terms, this becomes

1 877G6H = —(l—6 .2 3,01, p

Substituting for H), and 6H in eq. (5.5), it becomes

5: (/67rGp;,[w — 1]6.

Using the transformation

y= ln6,

eq. (5.12) becomes

3] = (/67rGp),[w -1].

(5.9)

(5.11)

(5.12)

(5.13)

(5.14)

This is a Langevin equation. Since it has another stochastic variable (pb), we have

to write the stochastic equation for p), also, and the resulting system of equations

has to be treated together, which leads to a two variable Fokker-Planck problem

and a two variable PDF. For this we define a transformation

89

11: = (5.15)Therefore from eq. (5.14) and the conservation equation for density, we have

y = —§ + (5.15)and

:'r = 1 + w(t). (5.17)To write the FPE, we need the drift and diffusion coeflicients. From the general

equation [82]

ii,‘ = hi + g,-J-1",-,the drift coeflicients are

8

Din = hi + Qkjt-92:91)‘, (5-19)and the diffusion terms are

D5? = gik g,-k. (5.20)From eqs. (5.16) and (5.17) the drift terms are obtained:

D9) = 1 (5.21)where we have used the fact that in the early universe pb is very high. The

diffusion terms are

90

1

D5? = p

D152) = Dry _ 1

D53 = D. (5.22)Here the diffusion terms arise due to fluctuations in the mean to factor alone, and

D is a constant. Now we write the FPE for the distribution function W(y, z, t):

7- +2aw 8D§” aD§,1> 020$} 6213;? 6 a[‘ ay ’ 69$ + ay2 632 53%DW1 W(y, 13,15). (5.23)

Substituting for the drift and diffusion coefficients

2 2 26W_[16 a 13 0 2a (524)E - ‘Ea’ 6-3.-+Fa—;fi+Da_z3+Ea3gE] WW‘)­

Neglecting the crossed term (to obtain an approximate solution) and applying a

separation ansatz,

W(y,:r,t) = u(1:)v(y)exp(——)\t) (5.25)

we get by variable separation

ldgv 1dv_ 2 ldu Dfl— — - 5.26vdyz vdy I uda: ud::2 ( )Here both sides can be equated to a constant (say c). We have a set of equations

for u(a:) and v(y). They are

—2 — — — ct) = 0, (5.27)

and

d2u 1 du 1 c— — —_ _ /\ _] = . .dz? Dd:r+D l +:::2 U 0 (528)Due to the essential singularity of the eq. (5.28) at :5 = 0, a series solution is

possible. However, if we take c = 0, then we will get a compact solution (in order

to understand the general behaviour). After solving equations (5.27) and (5.28),

we can write the complete solution as

W(y,:c, 2:) = Aexp y + % +1‘/ca: — /\t , (5.29)

where

/\ 12 j :- —— :mk _ D my (5.30)Physically reasonable solution exists for A 2 5. A is the normalisation constant,

chosen as 1/271'. The general solution is obtained by integrating eq. (5.29) in the

range —oo 3 k 3 00, and we get

W(y,:c,t) = e” exp [— ($4_D:)2] (5.31)7rDt

This is Gaussian in 1:. We can write the distribution function in terms of the

original variables 6 and pb (y = ln6 and z = 1/\/67rl§pb):

W(5,pb,t) = .:L_ l_L_ "677 % (532)/967r2GDp.'gt exp 247l’G'pg,Dt

This characterizes the non-deterministic (stochastic) evolution of density per­

turbations in the linear approximation and such an analysis becomes important

when we consider the expansion rate to be fluctuating.

92

5.3 Discussions and conclusions

When the universe is approximated by a many component fluid, the fluctuations

in creation rates are certainly physical processes which can lead to stochastic fluc­

tuations in the mean w factor of the equation of state (ie., a stochastic equation of

state). The evolution of the universe becomes stochastic (or the expansion rate of

the universe fiuctuates), where the time-evolution of the cosmological parameters

is described by the Langevin equations or SDES. We argue that such dynamical

processes may lead to a fractal distribution (or a scale invariant inhomogeneous

distribution) of galaxies, since the fluctuations in the evolution process never

lead to structures with perfect symmetry and most natural fractals were formed

through stochastic processes.

For a homogeneous distribution of galaxies, Hubble’s count law is [1]

N(< m) on 10°-20"", (5.33)where m is the apparent magnitude of the object and N represents number of

galaxies brighter than the magnitude m. D1 = 3 corresponds to standard model.

Equivalently, one can express (5.33) in terms of redshift (2) also, ie., N (< 2).

The apparent magnitude is related to z in the following way

D

m — M = 5log<1M;c) + 25, (5.34)where M is the absolute magnitude of the galaxy. The luminosity distance DL

is 1",-a(t0)[1 + 2]. In flat FRW models, 7*]-is calculated from

2./3to dt t, to dt _Tj-=‘[t' @= a(t-7-)Integrating

93

7-, = a(tj)Hj [\/1 + z - 1] , (5.36)and

DL = %(1 + 2)? [\/1 + z — 1], (5.37)where H (t) corresponds to the epoch tj. Thus the luminosity distance is related

to redshift, which depends on H (t). Fluctuations in the number counts around

the average behaviour as a function of m or 2 can discriminate between a genuine

fractal distribution and a homogeneous one [40]. Number counts versus apparent

magnitude can be used to test whether the large scale distribution of galaxies

(or clusters) can be compatible with a fractal or with a homogeneous behaviour

[41, 42]. In a fractal distribution, one expects to find persistent scale invariant

fluctuations around the average behaviour, which do not decay with m or z. On

the other hand, in a homogeneous distribution, on large enough scales, the relative

variance of the counts should decrease exponentially with m [1, 40]. Labini et

al., [40, 41] claim that, the relative fluctuations in the counts as a function of m

has a constant magnitude (for z 2 0.1), which can not be due to any smooth

correction to the data as evolution effects, but they can be the outcome of an

inhomogeneous distribution of galaxies.

In Chapter 2, we have shown that a fluctuating in factor, in turn, will lead

to fluctuations in the time-evolution of the Hubble parameter; ie., the expansion

rate of the universe becomes a stochastic quantity. We argue that such a fluctu­

ating expansion rate might have led to a randomness in the recession velocities

of objects, in addition to peculiar velocities (612 oc Q8'66p) induced by density

inhomogeneities. We also argue that, this randomness in the recession velocities

of galaxies led to an inhomogeneous (fractal) distribution of galaxies and clusters

94

of galaxies, since the dynamical equations containing stochastic quantities de­

scribe fractal growth [85, 92]. From (5.33) and (5.37), a fluctuating or stochastic

expansion rate (H (t)) may also provide the constant fluctuations observed in the

number count versus m (or z) relation.

Thus a stochastic evolution of the universe may provide the dynamical process

leading to the self-similar structures observed in the universe and also produce the

scatter observed in the Hubble diagram at high redshifts, where peculiar velocities

of standard model are inadequate. The present fluctuations found in the number

count versus apparent magnitude relation, which is a characteristic of fractal

distribution of galaxies, may also be due to the stochastic nature of the Hubble

parameter (See eqs. (5.34) and (5.37)). Since both density and Hubble parameter

are stochastic in the early epochs, the time—evolution of density perturbation

also must be non-deterministic, and hence described by SDES, as we have done

in section 5.2. In this chapter we have not made any attempt to characterize

the statistically scale invariant structures observed in the range of scales 1 to

20Oh‘1Mpc‘1 by measuring either the correlation function or power spectrum, we

have attempted only to provide a possible stochastic process which can produce

the observed inhomogeneous distribution of galaxies. The correlation function for

galactic distribution §(r) oc F”, for 7' < 10 h‘1Mpc is well established [15, 16].

However such statistical methods are based on the assumption of homogeneity

and hence are not appropriate to test the scale invariance of structures for a large

range of scales [36, 42]. Here we have only argued that a stochastic evolution of

the universe may lead to a scale invariant large scale structure.

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